Expected Value Calculator
Calculate the expected value (weighted average of outcomes) of a discrete probability distribution.
Expected Value E(X)
27.0000
Expected Value E(X) vs Outcome 1
Formula
## How to Calculate Expected Value ### Formula **E(X) = Sum of (xi * pi)** The expected value is the long-run average outcome of a random variable. Multiply each possible outcome by its probability and sum the products. It represents the "center of gravity" of the probability distribution. The variance is E(X^2) - [E(X)]^2.
Exemplo Resolvido
A game pays $10 with probability 0.2, $20 with probability 0.5, and $50 with probability 0.3.
- 01E(X) = 10*0.2 + 20*0.5 + 50*0.3
- 02= 2 + 10 + 15 = 27
- 03E(X^2) = 100*0.2 + 400*0.5 + 2500*0.3 = 20 + 200 + 750 = 970
- 04Var(X) = 970 - 27^2 = 970 - 729 = 241
- 05SD = sqrt(241) ≈ 15.5242
Perguntas Frequentes
Does the expected value have to be a possible outcome?
No. The expected value is a weighted average and often falls between outcomes. For example, the expected value of a fair die roll is 3.5, which is not a possible outcome.
What if probabilities do not sum to 1?
For a valid probability distribution, probabilities must sum to exactly 1. If they do not, either the distribution is incomplete or there is an error in the inputs.
How is expected value used in decision making?
Expected value helps compare options by their long-run average payoff. In finance, it estimates average returns. In gambling, a positive expected value means a profitable bet on average.
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