How Do You Calculate the Expected Value of a Discrete Random Variable?

To calculate the expected value of a discrete random variable, multiply each possible value by its probability and add the resulting products. In symbols, $E(X)=\sum_{i=1}^{n}x_iP(X=x_i)$.

The Method

The expected value is the probability-weighted mean of all possible values of a discrete random variable. Before calculating it, check that the probability distribution is valid: every probability must be between $0$ and $1$, and

$$\sum_{i=1}^{n}P(X=x_i)=1.$$

Then follow these steps:

1. Multiply each outcome $x_i$ by its corresponding probability $P(X=x_i)$.
2. Add all the products.
3. Interpret the result in the context of many repetitions of the experiment.

Because probabilities act as weights, outcomes with larger probabilities have more influence on $E(X)$. The calculation uses every possible outcome, unlike the ordinary mean of the listed values, which would incorrectly treat the outcomes as equally likely.

Worked Example

A game gives a player the following gains, where negative values represent losses.

First, confirm that $0.50+0.30+0.20=1$. Then calculate

$$E(X)=(-2)(0.50)+(1)(0.30)+(5)(0.20)=0.30.$$

The expected gain is therefore $0.30$ units per game. This does not mean the player will gain exactly $0.30$ in one game; it means the mean gain approaches $0.30$ per game over many repetitions.

Expected value also predicts an expected total over repeated trials. If the game is played $100$ independent times under the same conditions, the expected total gain is $100E(X)=100(0.30)=30$ units. This is still not a guaranteed result; random variation means the actual total may differ.

A common misconception is that the expected value must be one of the possible outcomes. It does not: it is a long-run average. In a game, $E(X)=0$ indicates a fair game, while a positive expected value favours the player.

IB Exam Technique

This is SL 4.7: Discrete random variables. Show the substitution into $E(X)=\sum_{i=1}^{n}x_iP(X=x_i)$, not only a GDC answer, and interpret the result in context when the command term requires it. Keep full precision until the final answer.