4.5.2 Expected Values

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 $X$ with its corresponding probability
- Adding all these terms together

$E (X) = \sum x P (X = x)$

- - 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

Worked example

Daphne pays $15 to play a game where she wins a prize of $1, $5, $10 or $100. The random variable $W$ represents the amount she wins and has the probability distribution shown in the following table: