The transformation theorem provides a straightforward means of computing the expected value of a function of a random variable, without requiring knowledge of the probability distribution of the function whose expected value we need to compute.

For discrete random variables, the theorem is as follows.

Proposition Let be a discrete random variable and a function. DefineThen,where is the support of and is its probability mass function.

Note that the above formula does not require knowledge of the support and the probability mass function of , unlike the standard formula

For continuous random variables, the theorem is as follows.

Proposition Let be a continuous random variable and a function. DefineThen,where is the probability density function of .

Similarly to what we said before, the above formula does not require knowledge of the probability density function of , unlike the standard formula

More details about the transformation theorem can be found in the lecture entitled Expected value.

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