E(X) & Var(X) of Discrete Random Variables

E(X) of DRVs

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

$E (X) = \sum x P (X = x)$

Look out for symmetrical distributions (where the values of $X$ are symmetrical and their probabilities are symmetrical)
- 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 is 5

Worked example

Daphne pays $15 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
$P (W = w)$	0.35	0.5	0.05	0.1

Calculate the expected value of Daphne's prize.

expected-values-we

Var(X) of DRVs

How do I calculate E(X²)?

$E (X^{2})$ means the expected value or the mean of the random variable $X^{2}$
It is calculated by:
- Squaring each value of $X$ to get the values of $X^{2}$
- Multiplying each value of $X^{2}$ by its corresponding probability
- Adding all these terms together
It's formula is $E (X) = \sum_{}^{} x^{2} P (X = x)$

Is E(X²) equal to (E(X))²?

No!
$E (X^{2})$ is the mean of the squares of $X$
${(E (X))}^{2}$ is the square of the mean of $X$
For example, if $X$ = 1 or $X$ = -1, both with probabilities of 0.5 then
- the mean is 0 so ${(E (X))}^{2}$ = 0² = 0
- but the squares of $X$ are 1 and 1, so $E (X^{2}) = 1$

What is Var(X) and how do I calculate Var(X)?

$Var (X)$ means the variance of a random variable $X$
- How spread out $X$ values are from their mean
  - High $Var (X)$ means more spread out
- How consistent $X$ values are around their mean
  - High $Var (X)$ means more variability so less consistent
For any random variable this can be calculated using the formula

$Var (X) = E (X^{2}) - (E {(X))}^{2}$

- This is the mean of the squares of $X$ minus the square of the mean of $X$

$Var (X)$ is always positive
The standard deviation of a random variable $X$ is $\sqrt{Var (X)}$
An alternative formula is $Var (X) = E [{(X - E (X))}^{2}]$
- The expected value of the squares of the distances from the mean
- This is its formal definition, but less useful in practice

How do I find E(X²) if Var(X) and E(X) are known?

Rearrange the formula $Var (X) = E (X^{2}) - {(E (X))}^{2}$
- $E (X^{2}) = Var (X) + {(E (X))}^{2}$
- Then substitute in the values for $Var (X)$ and $E (X)$

Examiner Tip

Check if your answer makes sense:
- The mean should fit within the range of the values of X.
- The variance must be positive.

Worked example

The discrete random variable $X$ has the probability distribution shown in the following table:

$x$	2	3	5	7
$P (X = x)$	0.1	0.3	0.2	0.4

(a)

Find the value of

E (X)

3-1-2-ex-_-varx-discrete-we-solution_a

(b)

Find the value of

E (X^{2})

3-1-2-ex-_-varx-discrete-we-solution_b

(c)

Find the value of

Var (X)

3-1-2-ex-_-varx-discrete-we-solution_c

E(g(X)) of DRVs

How do I calculate E(g(X))?

$E (X^{2})$ means $\sum_{}^{} x^{2} P (X = x)$
- The expected value (mean) of $X^{2}$
- $X^{2}$ is a function of $X$
Let $g (X)$ be any function of $X$
- Then $E (g (X)) = \sum_{}^{} g (x) P (X = x)$
- Multiply values of $g (x)$ by their corresponding probabilities
$E (g (X))$ does not mean substitute the value of $E (X)$ into $g (x)$
- $E (g (X)) \neq g (E (X))$

3-1-2-ex-_-varx-discrete-diagram-2

Worked example

The random variable $X$ has the following probability distribution.

$x$	1	8	27
$P (X = x)$	0.1	0.3	0.6

Determine which out of $E (\sqrt[3]{X})$ or $E (\ln X)$ is greater.

egx-of-drvs

E(X) & Var(X) of Discrete Random Variables (Edexcel A Level Further Maths: Further Statistics 1)

Revision Note

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

How do I calculate E(X2)?

Is E(X²) equal to (E(X))²?

What is Var(X) and how do I calculate Var(X)?

How do I find E(X2) if Var(X) and E(X) are known?

How do I calculate E(g(X))?

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Discrete Probability Distributions

Discrete Probability Distributions

Linear Transformations of DRVs

Problem Solving with DRVs

Standard Discrete Models

Poisson & Binomial Distributions

The Poisson Distribution

Poisson Approximations of Binomials

Geometric & Negative Binomial Distributions

The Geometric Distribution

The Negative Binomial Distribution

Probability Generating Functions

Probability Generating Functions

Probability Generating Functions (PGFs)

E(X) & Var(X) of PGFs

PGFs of Standard Distributions

PGFs of Sums & Transformations

Central Limit Theorem

Central Limit Theorem

Central Limit Theorem

Hypothesis Testing

Poisson & Geometric Hypothesis Testing

Poisson Hypothesis Testing

Geometric Hypothesis Testing

Chi Squared Tests

Goodness of Fit

Chi Squared Tests for Standard Distributions

Chi Squared Tests for Contingency Tables

Quality of Tests

Type I & Type II Errors

Size & Power of Test

Author: Mark

How do I calculate E(X²)?

How do I find E(X²) if Var(X) and E(X) are known?