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

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

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 

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
  •