Functions of random variables

Let $x$ be some random variable. Then $y = f(x)$ is a function of that random variable. Given the probability distribution of the variable $p(x)$ , we wish to find the distribution of the function $p(y)$ .

Consider first that we can write the probability density at some given $x_0$ as an integral over the entire range of $x$

p(x_0) = \int_{-\infty}^\infty p(x) \delta(x-x_0) \dx.

We can equivalently write the probability of finding some $y=y_0$ as an integral over the range of $x$ , and selecting the values where $f(x)=y_0$ using a delta function.

p(y_0) = \int_{-\infty}^\infty p(x) \delta(y_0-f(x)) \dx.

Evaluating this integral requires us to recall that integrating a delta applied to a function yields a result inversely proportional to that function’s derivative.

\int f(x) \delta(g(x)) \dx = \frac{f(x_0)}{|g'(x_0)|}.

Thus, allowing $x_i$ to be the roots of $f(x)$ , we find

p(y_0) = \sum_{f(x_i)=0} \frac{p(x_i)}{|f'(x_i)|}.

For a function of multiple variables, write the probability similarly as several integrals of the joint probability density.