$f(x)$ and $F(x)$ are the pdf and cdf of a distribution such that f'(x) exists for all $x$. Let the distribution $g(y) = f(y)/F(b)$ for $-\infty < y < b$ have mean $-f(b)/F(b)$ for all real $b$. Prove that $f(x)$ is a pdf of a standard normal distribution.
Attempt: I get $-f(b)= \int_{-\infty}^b y f(y) \,dy$ by definition of expected value. I am stuck here.