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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 4. Statistics & Probability / 4.4 Probability Distributions / 4.4.2 Mean & Variance


4.4.2 Mean & Variance


Expected Values E(X)

What does E(X) mean and how do I calculate E(X)?

  • E(X) means the expected value or the mean of a random variable X
    • The expected value does not need to be an obtainable value of X
    • For example: the expected value number of times a coin will land on tails when flipped 5 times is 2.5
  • For a discrete random variable, it is calculated by:
    • Multiplying each value of X with its corresponding probability
    • Adding all these terms together

begin mathsize 22px style straight E left parenthesis X right parenthesis equals sum x straight P left parenthesis X equals x right parenthesis end style

      • This is given in the formula booklet
  • Look out for symmetrical distributions (where the values of X are symmetrical and their probabilities are symmetrical) as the mean of these is the same as the median
    • For example: if X can take the values 1, 5, 9 with probabilities 0.3, 0.4, 0.3 respectively then by symmetry the mean would be 5

How can I decide if a game is fair?

  • Let X be the random variable that represents the gain/loss of a player in a game
    • X will be negative if there is a loss
  • Normally the expected gain or loss is calculated by subtracting the cost to play the game from the expected value of the prize
  • If E(X) is positive then it means the player can expect to make a gain
  • If E(X) is negative then it means the player can expect to make a loss
  • The game is called fair if the expected gain is 0
    • E(X) = 0

Worked Example

Daphne pays $5 to play a game where she wins a prize of $1, $5, $10 or $100. The random variable W represents the amount she wins and has the probability distribution shown in the following table:

w 1 5 10

100
straight P left parenthesis W equals w right parenthesis 0.35 0.5 0.05 0.01
a)
Calculate the expected value of Daphne's prize.

4-4-2-ib-ai-aa-sl-expected-values-a-we-solution

b)
Determine whether the game is fair.

4-4-2-ib-ai-aa-sl-expected-values-b-we-solution

Variance Var(X)

What does Var(X) mean and how do I calculate Var(X)?

  • Var(X) means the variance of a random variable X
    • The standard deviation is the square root of the variance
      • This provides a measure of the spread of the outcomes of X
    • The variance and standard deviation can never be negative
  • The variance of X is the mean of the squared difference between X and the mean

begin mathsize 22px style Var invisible function application open parentheses X close parentheses equals straight E invisible function application open parentheses X minus mu close parentheses squared end style

    • This is given in the formula booklet
  • This formula can be rearranged into the more useful form:

begin mathsize 22px style Var invisible function application open parentheses X close parentheses equals straight E invisible function application open parentheses X squared close parentheses minus left square bracket straight E invisible function application open parentheses X close parentheses right square bracket ² end style

    • This is given in the formula booklet
      • Compare this formula to the formula for the variance of a set of data
    • This formula works for both discrete and continuous X

How do I calculate E(X²) for discrete X?

  • E(X²) means the expected value or the mean of the random variable defined as X²
  • For a discrete random variable, it is calculated by:
    • Squaring each value of X to get the values of X2
    • Multiplying each value of X2 which its corresponding probability
    • Adding all these terms together
      • straight E invisible function application open parentheses X squared close parentheses equals sum x squared straight P open parentheses X equals x close parentheses
      • This is given in the formula booklet as part of the formula for Var(X)
      • Var invisible function application open parentheses X close parentheses equals sum x ² straight P invisible function application left parenthesis X equals x right parenthesis minus mu ²
  • E(f(X)) can be found in a similar way

Is E(X²) equal to E(X)²?

  • Definitely not!
    • They are only equal if X can only take one value
  • E(X²) is the mean of the values of X²
  • E(X)² is the square of the mean of the values of X
  • To see the difference
    • Imagine a random variable X that can only take 1 and -1 with equal chance
    • E(X) = 0 so E(X)² = 0
    • The square values are 1 and 1 so E(X²) = 1

Exam Tip

  • In an exam you can enter the probability distribution into your GDC using the statistics mode
    • Enter the possible values as the data
    • Enter the probabilities as the frequencies
  • You can then calculate the mean and variance just like you would with data

Worked Example

The score on a game is represented by the random variable S defined below.

s

0

1

2

10

straight P left parenthesis S equals s right parenthesis

0.4

0.3

0.25

0.05

Calculate Var left parenthesis S right parenthesis.

4-4-2-ib-aa-hl-variance-we-solution

Transformation of a Single Variable

How do I calculate the expected value and variance of a transformation of X?

  • Suppose X is transformed by the function f to form a new variable T = f(X)
    • This means the function f is applied to all possible values of X
  • Create a new probability distribution table
    • The top row contains the values t subscript i equals f open parentheses x subscript i close parentheses
    • The bottom row still contains the values straight P invisible function application open parentheses X equals x subscript i close parentheses which are unchanged as:
      • straight P invisible function application open parentheses X equals x subscript i close parentheses equals straight P invisible function application open parentheses f open parentheses X close parentheses equals f open parentheses x subscript i close parentheses close parentheses equals straight P invisible function application left parenthesis T equals t subscript i right parenthesis
      • Some values of T may be equal so you can add their probabilities together
  • The mean is calculated in the same way
    • straight E invisible function application open parentheses T close parentheses equals sum t straight P invisible function application left parenthesis X equals x right parenthesis blank
  • The variance is calculated using the same formula
    • Var invisible function application open parentheses T close parentheses equals straight E invisible function application open parentheses T ² close parentheses minus open square brackets straight E invisible function application open parentheses T close parentheses close square brackets squared

Are there any shortcuts?

  • There are formulae which can be used if the transformation is linear
    • T equals a X plus b where a and b are constants
  • If the transformation is not linear then there are no shortcuts
    • You will have to first find the probability distribution of T

What are the formulae for E(aX ± b) and Var(aX ± b)?

  • If a and b are constants then the following formulae are true:
    • E(aX ± b) = aE(X) ± b
    • Var(aX ± b) = a² Var(X)
      • These are given in the formula booklet
  • This is the same as linear transformations of data
    • The mean is affected by multiplication and addition/subtraction
    • The variance is affected by multiplication but not addition/subtraction
  • Remember division can be written as a multiplication
    • X over a equals 1 over a X

Worked Example

X is a random variable such that straight E left parenthesis X right parenthesis equals 5and Var left parenthesis X right parenthesis equals 4.

Find the value of:

(i)
straight E left parenthesis 3 X plus 5 right parenthesis
(ii)
Var left parenthesis 3 X plus 5 right parenthesis
(iii)
Var left parenthesis 2 minus X right parenthesis.

4-4-2-ib-aa-ai-hl-axb-we-solution



  • 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
                                                                                                                                                                                                                                                                                                        Daniel Finlay

                                                                                                                                                                                                                                                                                                        Author: Daniel

                                                                                                                                                                                                                                                                                                        Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.


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