4.5.2 Expected Values

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