Mathematical expectation or Expected value
The expected value is a weighted average
of the values of a random variable may assume.
The weights are the probabilities.
Let X be a discrete random variable with probability mass function (p.m.f.) p(x) .
Then, its expected value is defined by
If X is a continuous random variable and f(x) is the value of its probability density function at x, the expected value of X is
- E(X) is defined to be the indicated series provided that the series is absolutely convergent; otherwise, we say that the mean does not exist.
- In (1), E(X) is an “average” of the values that the random variable takes on, where each value is weighted by the probability that the random variable is equal to that value. Values that are more probable receive more weight
- In (2), E(X) is defined to be the indicated integral if the integral exists; otherwise, we say that themean does not exist
- In (2), E(X) is an “average” of the values that the random variable takes on, where each value x ismultiplied by the approximate probability that X equals the value x, namely fₓ(x)dx and then integrated over all values.
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