Statistics / Unit 7: Random Variables
7.2 Expected Value
Notes
The expected value (mean) of a discrete random variable is $\displaystyle E(X) = \mu_X = \sum x \cdot P(x)$.
- It is the long-run average over many repetitions. It need not be a possible value of $X$.
- Multiply each value by its probability, then add.
- Example: $X = 0, 1, 2$ with probabilities $0.2, 0.5, 0.3$ gives $E(X) = 0(0.2) + 1(0.5) + 2(0.3) = 1.1$.
- A game is fair when the expected net gain is 0.
Practice
In a game you win $10 with probability $0.1$ and $0 otherwise. What is the expected winnings per play?
$10(0.1) + 0(0.9) = 1$.
$X = 1, 2, 3$ with probabilities $0.5, 0.3, 0.2$. What is $E(X)$?
$1(0.5) + 2(0.3) + 3(0.2) = 0.5 + 0.6 + 0.6 = 1.7$.
A raffle sells 1,000 tickets at $2 each. One ticket wins a $1,000 prize. What is the expected net gain for one ticket?
Expected prize is $\displaystyle \frac{1000}{1000} = 1$, and the cost is 2, so the net is $1 - 2 = -1$ dollar.
Ready for more? Take the Unit 7 test.