Mean & Variance (DP IB Analysis & Approaches (AA)): Revision Note

Expected Values E(X)

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

Variance Var(X)

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 with 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

Transformation of a Single Variable

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:

  • 

  • 

    • 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

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