Equations · Probability and statistics
Expected value
The long-run average of a random quantity: each possible value weighted by its probability.
E[X] = ∑ᵢ xᵢ P(X = xᵢ)
Aethel says
the expected value of X, equals the sum over i of x sub i times the probability that X equals x sub i
What it says
Multiply each value the random variable X can take by the probability that it takes it, and add the results. For a fair die, that is 3.5, even though no single roll shows 3.5.
Averages of many independent repetitions get closer and closer to the expected value: the law of large numbers.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| E[X] | the expected value of X | the mean of the random variable X |
| xᵢ | x sub i | the ith possible value |
| P(X = xᵢ) | the probability that X equals x sub i | how likely that value is |
More on∑Summation (sigma notation)
Where you meet it
Probability, statistics, decision theory, finance and reinforcement learning.
Questions
How do you say Expected value out loud?
E[X] = ∑ᵢ xᵢ P(X = xᵢ) is read: “the expected value of X, equals the sum over i of x sub i times the probability that X equals x sub i.” That is exactly what Aethel says when it meets the formula in a book.
What does E[X] mean in Expected value?
E[X] (said “the expected value of X”) is the mean of the random variable X.
What does xᵢ mean in Expected value?
xᵢ (said “x sub i”) is the ith possible value.
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