4.4.2 Mean & Variance

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 with its corresponding probability
  • Adding all these terms together

• • • This is given in the formula booklet
• Look out for symmetrical distributions (where the values of X are symmetrical and their probabilities are symmetrical) as 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 would be 5

How can I decide if a game is fair?

• Let X be the random variable that represents the gain/loss of a player in a game
  • X will be negative if there is a loss
• Normally the expected gain or loss is calculated by subtracting the cost to play the game from the expected value of the prize
• If E(X) is positive then it means the player can expect to make a gain
• If E(X) is negative then it means the player can expect to make a loss
• The game is called fair if the expected gain is 0
  • E(X) = 0

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

• Var(X) means the variance of a random variable X
  • The standard deviation is the square root of the variance
    • This provides a measure of the spread of the outcomes of X
  • The variance and standard deviation can never be negative
• The variance of X is the mean of the squared difference between X and the mean

• • This is given in the formula booklet
• This formula can be rearranged into the more useful form:

• This is given in the formula booklet
• Compare this formula to the formula for the variance of a set of data
• This formula works for both discrete and continuous X

How do I calculate E(X²) for discrete X?

• E(X²) means the expected value or the mean of the random variable defined as X²
• For a discrete random variable, it is calculated by:
  • Squaring each value of X to get the values of X²
  • Multiplying each value of X² which its corresponding probability
  • Adding all these terms together
    • 
    • This is given in the formula booklet as part of the formula for Var(X)
    • 
• E(f(X)) can be found in a similar way

Is E(X²) equal to E(X)²?

• Definitely not!
• They are only equal if X can only take one value
• E(X²) is the mean of the values of X²
• E(X)² is the square of the mean of the values of X
• To see the difference
• Imagine a random variable X that can only take 1 and -1 with equal chance
• E(X) = 0 so E(X)² = 0
• The square values are 1 and 1 so E(X²) = 1

How do I calculate the expected value and variance of a transformation of X?

• Suppose X is transformed by the function f to form a new variable T = f(X)
  • This means the function f is applied to all possible values of X
• Create a new probability distribution table
  • The top row contains the values 
  • The bottom row still contains the values  which are unchanged as:
    • 
    • Some values of T may be equal so you can add their probabilities together
• The mean is calculated in the same way
  • 
• The variance is calculated using the same formula
  • 

Are there any shortcuts?

• There are formulae which can be used if the transformation is linear
•  where a and b are constants
• If the transformation is not linear then there are no shortcuts
• You will have to first find the probability distribution of T

What are the formulae for E(aX ± b) and Var(aX ± b)?

• If a and b are constants then the following formulae are true:
  • E(aX ± b) = aE(X) ± b
  • Var(aX ± b) = a² Var(X)
    • These are given in the formula booklet
• This is the same as linear transformations of data
  • The mean is affected by multiplication and addition/subtraction
  • The variance is affected by multiplication but not addition/subtraction
• Remember division can be written as a multiplication
  •