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Transformation theorem

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.

Transformation theorem for discrete random variables

For discrete random variables, the theorem is as follows.

Proposition Let X be a discrete random variable and [eq1] a function. Define[eq2]Then,[eq3]where R_X is the support of X and [eq4] is its probability mass function.

Note that the above formula does not require knowledge of the support and the probability mass function of $Y,$, unlike the standard formula[eq5]

Transformation theorem for continuous random variables

For continuous random variables, the theorem is as follows.

Proposition Let X be a continuous random variable and [eq6] a function. Define[eq2]Then,[eq8]where [eq9] is the probability density function of X.

Similarly to what we said before, the above formula does not require knowledge of the probability density function of $Y,$, unlike the standard formula[eq10]

More details

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

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