Expected value
The expected value E(X) is the long-run average of a random variable. Multiply each value by its probability and add the results. It need not be a value that can actually occur.
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Analogy
If you played a game thousands of times, the expected value is your average result per play.
Example
x: 0 1 2P(x): 0.5 0.3 0.2E(X) = 0(0.5) + 1(0.3) + 2(0.2)= 0 + 0.3 + 0.4 = 0.7
In practice
Casinos and insurers price their products so the expected value favours them.
Practice
Flashcards
- How do you find E(X) from a table?Multiply each value by its probability and add.
Interview questions
- A game costs 1 to play. You roll a fair die and win 6 if it shows a six, otherwise you win nothing. Find the expected profit and say whether the game is fair.
- Expected winnings = 6 x 1/6 = 1
- Expected profit = 1 - 1 = 0
- Game is fair
- Expected value is a long-run average
Questions people ask
Does the expected value have to be a possible outcome?
No. A fair die has expected value 3.5, yet no roll gives 3.5. The expected value is a long-run average, not a prediction for one trial. Think of it as the balance point of the distribution rather than a likely result.
How is expected value different from the mean of a sample?
The expected value is a fixed property of the whole distribution, worked out from probabilities. A sample mean is calculated from observed data and changes from sample to sample. By the law of large numbers, sample means settle near the expected value as samples grow.