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

E(X) of DRVs

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

•  means the expected value or the mean of a random variable 

  • The expected value does not need to be an obtainable value of

  • 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 with its corresponding probability

  • Adding all these terms together

• Look out for symmetrical distributions (where the values of are symmetrical and their probabilities are symmetrical)

  • The mean of these is the same as the median

  • For example: if  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)?

•  means the expected value or the mean of the random variable 

• It is calculated by:

  • Squaring each value of to get the values of 

  • Multiplying each value of  by its corresponding probability

  • Adding all these terms together

• It's formula is

Is E(X²) equal to (E(X))²?

• No!

•  is the mean of the squares of 

•  is the square of the mean of 

• For example, if  = 1 or  = -1, both with probabilities of 0.5 then

  • the mean is 0 so  = 0² = 0

  • but the squares of are 1 and 1, so

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

•  means the variance of a random variable 

  • How spread out  values are from their mean

    • High  means more spread out

  • How consistent  values are around their mean

    • High means more variability so less consistent

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

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

•  is always positive

• The standard deviation of a random variable  is 

• An alternative formula is

  • 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 

  • 

  • Then substitute in the values for and 

E(g(X)) of DRVs

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

•  means  

  • The expected value (mean) of 

  • is a function of

• Let  be any function of 

  • Then

  • Multiply values of  by their corresponding probabilities

• does not mean substitute the value of into

  •