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Some Notes about Expected Values

Posted in Math, Statistics by Sina Iravanian on August 21, 2011

Expected value of a continuous random variable is given by:

\mathbb{E}[X] = \int_{-\infty}^{+\infty} x\,f(x)\,dx

where f is the probability density function of the random variable x. Now the question is how do we calculate \mathbb{E}[g(X)], e.g., \mathbb{E}[X^2]? Do we know f(g(x)) for x \in X? The answer is that we don’t need to. No matter what we do with x \in X, by applying g to it, we have:

f(g(x)) = f(x)

therefore:

\mathbb{E}[g(X)] = \int_{-\infty}^{+\infty} g(x)\,f(x)\,dx.

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