Let $X$ be a normally-distributed random variable with mean $0$ and some positive variance.
Denote $f$ as its probability density function, and $F$ as its cumulative distribution function.
I am interested in calculating
$$\int_{-\infty}^{0}F(kx)f(x)\text{ d}x$$
where $k \neq 0$ is a known constant that does not depend on $x$.
How exactly would I calculate this?
When I saw this integral, I made the mistake of thinking it was $\mathbb{E}[F(kX)]$, but soon realized it wasn't, as the integral does not cover all of $\mathbb{R}$, but only $(-\infty, 0)$.
No complete solutions are necessary; hints are appreciated.
Best Answer
Hint: Write it as $$ \int_{-\infty}^0 F(kx)f(x)\,\mathrm{d}x =\iint_{\mathbb{R}^2}f(x)f(y)\mathbf{1}_{x<0}\mathbf{1}_{y<kx}\,\mathrm{d}\mathcal{L}^2(x,y) $$ and $\{x<0, y<kx\}$ can be easily described in polars.