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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.

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 Tips and Tricks

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 $E (X)$ .

(b) Find the value of $E (X^{2})$ .

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 $g (x)$
- $E (g (X)) \neq g (E (X))$

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.

Further Maths: Further Statistics 1

ex-and-varx-of-discrete-random-variables

E(X) of DRVs

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

Worked Example

Var(X) of DRVs

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?

Examiner Tips and Tricks

Worked Example

E(g(X)) of DRVs

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

Worked Example

Responses

Further Maths: Further Statistics 1

ex-and-varx-of-discrete-random-variables

E(X) of DRVs

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

Var(X) of DRVs

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?

E(g(X)) of DRVs

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

Responses

How do I calculate E(X²)?

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