I was going through a problem in Geoffrey Grimmett and David Stirzaker's book (Probability and Random Processes). The problem is as follows:
If $f$ and $g$ are probability density functions, then prove that for $ 0 \leq \lambda \leq 1$ the function $\lambda f + (1-\lambda)g$ is a density function. Is the product $fg$ a density function as well?
It is straightforward to prove $\lambda f + (1-\lambda)g$ is a density function. For the second question as well, one can construct trivial functions for $f$ and $g$ as $f(x)=g(x)=1$ for $ 0 \leq x \leq 1$.
Are there any other non-trivial examples of a family or class of distributions for which one can find $\int_{-\infty}^{\infty} f(x)g(x) dx=1$?
Best Answer
Given any probability densities $f(x)$ and $g(x)$ with $f(x) g(x) > 0$ on a set of positive measure, and any constant $r > 0$, $r f(rx)$ and $r g(rx)$ are also probability densities, and $$\int_{\mathbb R} (r f(rx))(r g(rx))\ dx = r \int_{\mathbb R} f(x) g(x)\ dx$$ We can then choose $r$ so that this is $1$. That gives us two probability densities $r f(rx)$ and $r g(rx)$ whose product is a probability density.