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
Daphne pays $5 to play a game where she wins a prize of $1, $5, $10 or $100. The random variable represents the amount she wins and has the probability distribution shown in the following table:
Calculate the expected value of Daphne's prize.
Determine whether the game is fair.