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

E(X) of DRVs

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

• $\mathrm{E}\left(X\right)$ 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

$\mathrm{E}\left(X\right)=\sum x\mathrm{P}(X=x)$

• 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

Var(X) of DRVs

How do I calculate E(X²)?

• $\mathrm{E}\left(X^2\right)$ 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

Is E(X²) equal to (E(X))²?

• No!

• $\mathrm{E}\left(X^2\right)$ is the mean of the squares of $X$

• ${\left(\mathrm{E}\left(X\right)\right)}^2$ 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 ${\left(\mathrm{E}\left(X\right)\right)}^2$ = 0² = 0

  • but the squares of $X$ are 1 and 1, so $\mathrm{E}\left(X^2\right)=1$

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

• $\mathrm{Var}\left(X\right)$ means the variance of a random variable $X$

  • How spread out $X$ values are from their mean

    • High $\mathrm{Var}\left(X\right)$ means more spread out

  • How consistent $X$ values are around their mean

    • High $\mathrm{Var}\left(X\right)$ means more variability so less consistent

• For any random variable this can be calculated using the formula

$\mathrm{Var}\left(X\right)=\mathrm{E}\left(X^2\right)-(\mathrm{E}{\left(X\right))}^2$

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

• $\mathrm{Var}\left(X\right)$ is always positive

• The standard deviation of a random variable $X$ is $\sqrt{\mathrm{Var}\left(X\right)}$

• An alternative formula is $\mathrm{Var}\left(X\right)=\mathrm{E}\left[{\left(X-\mathrm{E}\left(X\right)\right)}^2\right]$

  • 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 $\mathrm{Var}\left(X\right)=\mathrm{E}\left(X^2\right)-{\left(\mathrm{E}\left(X\right)\right)}^2$

  • $\mathrm{E}\left(X^2\right)=\mathrm{Var}\left(X\right)+{\left(\mathrm{E}\left(X\right)\right)}^2$

  • Then substitute in the values for $\mathrm{Var}\left(X\right)$ and $\mathrm{E}\left(X\right)$

E(g(X)) of DRVs

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

• $\mathrm{E}\left(X^2\right)$ means  

  • The expected value (mean) of $X^2$

  • $X^2$ is a function of $X$

• Let $\mathrm{g}\left(X\right)$ be any function of $X$

  • Then

  • Multiply values of $\mathrm{g}\left(x\right)$ by their corresponding probabilities

• $\mathrm{E}\left(\mathrm{g}\left(X\right)\right)$ does not mean substitute the value of $\mathrm{E}\left(X\right)$ into $\mathrm{g}\left(x\right)$

  • $\mathrm{E}\left(\mathrm{g}\left(X\right)\right)\ne \mathrm{g}\left(\mathrm{E}\left(X\right)\right)$