Working through some problems in Introduction to Probability (Blitzstein).
Let F be the CDF of a continuous r.v., and f = F' be the PDF
(a) Show that g defined by g(x) = 2F(x)f(x) is also a valid PDF.
Does this have something to do with the fact that a PDF should integrate to 1?
(b) Show that h defined by $h(x) = \frac{1}{2}f(-x) + \frac{1}{2}f(x)$ is also a valid PDF.
Any hints on how to manipulate functions & their derivatives in ways that is useful for PDFs & CDFs? Are there some general concepts anyone knows?
Hints
on a)
If $F(x)$ is a CDF then so is $F(x)^2$.
What will be the looks of an eventual PDF for $F(x)^2$ if $F(x)$ has a derivative?
on b)
A function $k:\mathbb R\to\mathbb R$ is a PDF if and only if it is nonnegative and integrable with $\int_{-\infty}^{\infty} k(x)dx=1$.
If $f$ indeed has these qualities, then what can be said about $h(x)=\frac12f(-x)+\frac12f(x)$?