On this page we will explore the concept of discrete random variables and their probability distributions. We will look at how we can display probability distributions in tables or as functions. The questions are not so complicated (it is just a way of formalising the language of probability), for example, it is important not to forget that probabilities still add up to 1!
On this page, you should learn about
- discrete random variables
- probability distributions
- mean (expected value)
- variance (HL only)
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Here is a quiz that practises the skills from this page
START QUIZ!
A unbiased coin is flipped 3 times. The outcomes are listed below
HHH , HHT , HTH , THH , HTT , THT , TTH , TTT
Complete the probability distribution below for the random variable X, where X = number of heads flipped
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) |
There are 8 outcomes
TTT
P(X = 0) = 1/8
HTT , THT , TTH
P(X = 1) = 3/8
HHT , HTH , THH
P(X = 2) = 3/8
HHH
P(X = 3) = 1/8
A discrete random variable X has the following probability distribution
P(X = x) = \(\frac{1}{20}x(x + 1)\) , where x is 0 , 1 , 2 , 3
Complete the table below
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) |
P(X = 0) = \(\frac{1}{20}\cdot0(0 + 1)\)
P(X = 1) = \(\frac{1}{20}\cdot1(1 + 1)\)
P(X = 2) = \(\frac{1}{20}\cdot2(2 + 1)\)
P(X = 3) = \(\frac{1}{20}\cdot3(3 + 1)\)
Which is the correct function on the interval \(x\in{0,1,2,3,4}\) for the following probability distribution
If we work out P(X=1) for each of the distributions we can see which has the correct probability
\(\frac{1}{40} (1²+1)=0.05\)
\(\frac{1}{15}2^{1-1}\approx0.667\)
\(\left( \begin{matrix} 4 \\ 1 \end{matrix} \right) 0.5^{ 1 }0.5^{ { 4-1 } }=4\times0.0625=0.25\)
\(\frac{1}{24}(2\times1+1)=0.125\)
The random variable X has the following probability distribution
Find the value of k
k =
All probabilities add up to 1
0.1 + 0.35 + k + 0.15 + 0.05 = 1
0.65 + k = 1
k = 0.35
The random variable X has probability function
Find P( 2 < X < 8)
Find P( 2 < X < 8) =
The probability function of a discrete random variable X is given by
P(X=x) = \(\frac{x^2}{k}\) for \(x\in{1,2,3,4}\)
Find the value of k
k =
The sum of all the probabilities is 1
P(X=1) = \(\frac{1^2}{k}\)
P(X=2) = \(\frac{2^2}{k}\)
P(X=3) = \(\frac{3^2}{k}\)
P(X=4) = \(\frac{4^2}{k}\)
\(\frac{1}{k}+\frac{4}{k}+\frac{9}{k}+\frac{16}{k}=1\)
\(\frac{39}{k}=1\)
k = 39
The random variable X has probability function
P(X>3) = \(\frac{a}{b}\)
Hint: work out the value of k first
P(X>3) = \(\frac{a}{b}\)
a =
b =
P(X = 1) = k
P(X = 2) = 2k
P(X = 3) = 3k
P(X = 4) = 3k
P(X = 5) = 4k
All probabilities add up to 1
k + 2k + 3k + 3k + 4k = 1
13k = 1
k = \(\frac{1}{13}\)
P(X > 3) = P(X = 4) + P(X = 5)
P(X > 3) = 3k + 4k = \(\frac{7}{13}\)
The random variable X has probability function
By working out q first, find E(X)
E(X) =
All probabilities add up to 1
E(X) = \(\sum { xP(X=x) } \)
E(X) = 1x0.2 + 2x0.3 + 4x0.3 + 7x0.1 + 11x0.1 = 3.8
The random variable X represents the outcome of a game.
Which of the follwoing games are FAIR?
Select all the possible correct answers
For a fair game, E(X) = 0
E(X) = -2x0.4 + (-2)x0.05 + 0x0.05 + 1x0.1 + 2x0.3 = -0.2
E(X) = -3x0.4 + 0x0.1 + 1x0.1 + 2x0.1 +3x0.3 = 0
Since the distribution is symmetrical about x = 0, E(X) = 0
E(X) = -5x0.2 + 0x0.2 + 1x0.2 + 2x0.2 + 3x0.2 = 0.2
The discrete random variable X has the following probability distribution
E(X) = 3.5
Work out a and b
a =
b =
\(\sum { P(X=x)=1 } \)
0.1 + a + 0.3 + b + 0.1 = 1
a + b = 0.5
E(X) = \(\sum { xP(X=x) } \)
0x0.1 + 1xa + 3x0.3 + 5xb + 9x0.1 = 3.5
a + 0.9 + 5b + 0.9 = 3.5
a + 5b = 1.7
Solve the simultaneous equations
a + b = 0.5
a + 5b = 1.7
a = 0.2
b = 0.3
The discrete random variable X has probability function
P(X = x) = k(16 – x²) for x = 0, 1, 2, 3
- Find the value of the constant k
- Find P(1 ≤ X < 3)
Hint
Full Solution
The random variable X has probability function
a) Find the value of k
b) Work out \(P(X\ge 2)\)
Hint
Full Solution
Two boxes each contain three cards.
The first box contains cards labelled 1,3 and 5
The second box contains cards labelled 2, 4 and 6.
In a game, a player draws one card at random from each box and his score, X, is the sum of the numbers on the two cards.
a) Complete the following probability distribution
b) Work out E(X)
Hint
Full Solution
The following table shows the probability distribution of a discrete random variable X
If X represents the return from a game. Find a and b if the game is fair.
Hint
Full Solution
How much of Discrete Random Variables have you understood?
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