If f(x) is a density function, are these density functions aswell?

Question

In an exercise I was asked to decide which of a series of expressions were probability density functions given that $f(x)$ and $g(x)$ were PDFs. I got why $f(x)\times g(x) $ or $f(x)+ g(x)$ are not denstiy functions, but could not demonstrate why $f(x-c)$ and $f(c-x)$ are density funcions, and most importantly, I dont get why $f(e^x)$ is not a density function.

Answer 1

• You have that $$\int_{\mathbb R}f(x\pm c)dx=\int_{\mathbb R}f(x)dx=1,$$ so obviously, they are density function.

• Take $f(x)=\boldsymbol 1_{[0,1]}(x)$ which is a density function.

$$\int_{\mathbb R}f(e^x)dx=\int_0^\infty \frac{f(y)}{y}dy=\int_0^1\frac{1}{y}dy=\infty.$$