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 exam formula booklet

• 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 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 positive if there is a gain, and 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

  • 

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 of X

  • i.e. of the squared difference between and

• This is given in the exam formula booklet

• This formula can be rearranged into the more useful form:

• This is also given in the exam 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

    • these are the same as the probabilities for

  • Adding all these terms together

    • 

  • This is given in the exam formula booklet as part of the formula for Var(X):

    • 

• E(f(X)), where f is a function of 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