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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 4. Statistics & Probability / 4.7 Further Probability Distributions / 4.7.1 Probability Density Function


4.7.1 Probability Density Function


Calculating Probabilities using PDF

A continuous random variable can take any value in an interval so is typically used when continuous quantities are involved (time, distance, weight, etc)

What is a probability density function (p.d.f.)?

  • For a continuous random variable, a function can be used to model probabilities
    • This function is called a probability density function (p.d.f.), denoted by f(x)
  • For f(x) to represent a p.d.f. the following conditions must apply
    • f(x) ≥ 0 for all values of x
    • The area under the graph of y = f(x) must total 1
  • In most problems, the domain of x is restricted to an interval, a ≤ X ≤ b say, with all values of x outside of the interval having f(x)=0

How do I find probabilities using a probability density function (p.d.f.)?

  • The probability that the continuous random variable X lies in the interval a ≤ X ≤ b, where X has the probability density function f(x), is given by

space straight P left parenthesis a less or equal than X less or equal than b right parenthesis equals integral subscript a superscript b f left parenthesis x right parenthesis space straight d x

    • P(a ≤ X ≤ b) = P(a < X < b)
      • For any continuous random variable (including the normal distribution) P(X = n) = 0
      • One way to think of this is that a = b in the integral above
  • For linear functions it can be easier to find the probability using the area of geometric shapes
    • Rectangles: A = bh
    • Triangles: A = ½(bh)
    • Trapezoids: A = ½(a+b)h

How do I determine whether a function is a pdf?

  • Some questions may ask for justification of the use of a given function for a probability density function
    • In such cases check that the function meets the two conditions
      • f(x) ≥ 0 for all values of x
      • total area under the graph is 1

How do I use a pdf to find probabilities?

STEP 1
Identify the probability density function, f(x) - this may be given as a graph, an equation or as a piecewise function
e.g.  space f left parenthesis x right parenthesis equals open curly brackets table row cell 0.02 x end cell cell 0 less or equal than x less or equal than 10 end cell row 0 otherwise end table close
Identify the limits of X for a particular problem
Remember that P(a ≤ X ≤ b) = P(a < X < b)

STEP 2
Sketch, or use your GDC to draw, the graph of y = f(x)
Look for basic shapes (rectangles, triangles and trapezoids) as finding these areas is easier without using integration
Look for symmetry in the graph that may make the problem easier
Break the area required into two or more parts if it makes the problem easier

STEP 3
Find the area(s) required using basic shapes or integration and answer the question

  • Trickier problems may involve finding a limit of the integral given its value
    • i.e. Find one of the boundaries in the domain of X, given the probability
      • e.g. Find the value of a given that P(0 ≤ X ≤ a) = 0.09

Worked Example

The continuous random variable, X, has probability density function.

a)
Show that f(x) can represent a probability density function.

4-7-1-ib-hl-aa-only-we1a-soltn-

b)
Find, both geometrically and using integration, P(0 ≤ X ≤ 2).
4-7-1-ib-hl-aa-only-we1b-soltn-
c)
Write down P(X = 3.2).
4-7-1-ib-hl-aa-only-we1c-soltn-

Median & Mode of a CRV

What is meant by the median of a continuous random variable?

  • The median, m, of a continuous random variable, X, with probability density function f(x) is defined as the value of X such that

  • Since P(X = m) = 0 this can also be written as 
  • IF the p.d.f. is symmetrical (i.e. the graph of y = f(x) is symmetrical) then the median will be halfway between the lower and upper limits of x
    • In such cases the graph of y = f(x) has axis of symmetry in the line x = m

How do I find the median of a continuous random variable?

  • The median, m, of a continuous random variable, X, with probability density function f(x) is defined as the value of X such that

or

  • The equation that should be used will depend on the information in the question
    • If the graph of y = f(x) is symmetrical, symmetry may be used to deduce the median
    • This may often be the case if f(x) is linear and the area under the graph is a basic shape such as a rectangle

How do I find the median of a continuous random variable with a piecewise p.d.f.?

  • For piecewise functions, the location of the median will determine which equation to use in order to find it
    • For example
      • if space f stretchy left parenthesis x stretchy right parenthesis equals stretchy left curly bracket table row cell 1 fifth x end cell cell 0 less or equal than x less or equal than 2 end cell row cell 2 over 15 open parentheses 5 minus x close parentheses end cell cell 2 less or equal than x less or equal than 5 end cell row 0 otherwise end table
      • then integral subscript 0 superscript 2 1 fifth x space straight d x equals 0.4 so the median must lie in the interval 2 ≤ x ≤ 5
      • so to find the median, m, solve space integral subscript 2 superscript m 2 over 15 left parenthesis 5 minus x right parenthesis space straight d x equals 0.1
        ('0.4 of the area' already used for 0 ≤ x ≤ 2)
      • Use a GDC to plot the function and evalutae integral(s)

What is meant by the mode of a continuous random variable?

  • The mode of a continuous random variable, X, with probability density function f(x) is the value of x that produces the greatest value of f(x)

How do I find the mode of a continuous random variable?

  • This will depend on the type of function f(x); the easiest way to find the mode is by considering the shape of the graph of y = f(x)
  • If the graph is a curve with a maximum point, the mode can be found by differentiating and solving f’(x) = 0
    • If there is more than one solution to f’(x) = 0 then further work may be needed in deducing the mode
      • There could be more than one mode
      • Look for valid values of x from the domain of the p.d.f.
      • Use the second derivative (f’’(x)) to deduce the nature of each stationary point
      • Check the values of f(x) at the lower and upper limits of x, one of these could be the maximum value f(x) reaches
  • If the graph of y = f(x) is symmetrical, symmetry may be used to deduce the mode
    • For a symmetrical p.d.f., median = mode = mean

Worked Example

The continuous random variable X has probability function f(x) defined as

size 16px space size 16px f begin mathsize 16px style stretchy left parenthesis x stretchy right parenthesis end style size 16px equals size 16px 1 over size 16px 64 begin mathsize 16px style stretchy left parenthesis 16 x minus x cubed stretchy right parenthesis end style  size 16px space size 16px 0 size 16px less or equal than size 16px x size 16px less or equal than size 16px 4

a)
Find the median of X, giving your answer to three significant figures.

4-7-1-ib-hl-aa-only-we2a-soltn

b)
Find the exact value of the mode of X.

4-7-1-ib-hl-aa-only-we2b-soltn

Mean & Variance of a CRV

What are the mean and variance of a continuous random variable?

  • E(X) is the expected value, or mean, of the continuous random variable X
    • E(X) can also be denoted by μ
  • Var(X) is the variance of the continuous random variable X
    • Var(X) can also be denoted by σ2
    • The standard deviation, σ, is the square root of the variance

How do I find the mean and variance of a continuous random variable?

  • The mean is given by

mu equals straight E left parenthesis X right parenthesis equals integral subscript negative infinity end subscript superscript infinity x f left parenthesis x right parenthesis space straight d x

    • This is given in the formula booklet
    • If the graph of y = f(x) has axis of symmetry, x = a, then E(X) = a
  • The variance is given by

where space straight E left parenthesis X squared right parenthesis equals integral subscript negative infinity end subscript superscript infinity x squared f left parenthesis x right parenthesis space straight d x
    • This is given in the formula booklet
    • Another version of the variance is given in the formula booklet

space Var left parenthesis x right parenthesis equals integral subscript negative infinity end subscript superscript infinity left parenthesis x minus mu right parenthesis squared space f left parenthesis x right parenthesis space straight d x equals integral subscript negative infinity end subscript superscript infinity x squared f left parenthesis x right parenthesis space straight d x minus mu squared

    • but the first version above is usually more practical for solving problems
  • Be careful about confusing E(X2) and [E(X)]2
    • space straight E left parenthesis X squared right parenthesis equals integral subscript negative infinity end subscript superscript infinity x squared f left parenthesis x right parenthesis space straight d x   "mean of the squares"
    • space open square brackets straight E left parenthesis X right parenthesis close square brackets squared equals open square brackets integral subscript negative infinity end subscript superscript infinity x f left parenthesis x right parenthesis space straight d x close square brackets squared              "mean of the squares"

How do I find the mean and variance of a linear transformation of a continuous random variable?

  • For the continuous random variable, X, with mean E(X) and variance Var(X) then

space straight E left parenthesis a X plus b right parenthesis equals a straight E left parenthesis X right parenthesis plus b

and

space Var left parenthesis a X plus b right parenthesis equals a squared Var left parenthesis X right parenthesis

Exam Tip

  • Using your GDC to draw the graph of y = f(x) can highlight any symmetrical properties which reduce the work involved in finding the mean and variance

Worked Example

A continuous random variable, X, is modelled by the probability distribution function, f(x), such that

a)
Find the mean of X.

4-7-1-ib-hl-aa-only-we3a-soltn

b)
Find standard deviation of X.

4-7-1-ib-hl-aa-only-we3b-soltn



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Paul Freeman

                                                                                                                                                                                                                                                                                                        Author: Paul

                                                                                                                                                                                                                                                                                                        Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.


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