Assume two independent normal random variables $ h \sim N(\mu_h , \sigma_h ^2)$ and $ s \sim N(\mu_s , \sigma_s^2)$ with p.d.f. $f_h(x)$ and $f_s(x)$ respectively.
I need to compute the following double integral which is the result of computing a conditional probability:
For any $a \in R$
$ \int_{-\infty}^\infty \int_{-\infty}^{a-y} f_h(y) f_s(x)dxdy$ or equivalently $ \int_{-\infty}^\infty f_h(y) \int_{-\infty}^{a-y} f_s(x)dxdy$
I have been trying to do something with the explicit pdf of a normal distribution, but Im not getting anywhere. Is there maybe a way to use the error function for the calculation?
Thank you
Your integral seems to be calculating $\mathbb P(h+s \le a)$
Since $h$ and $s$ independent normal random variables, their sum is normally distributed with mean $\mu_h+\mu_s$ and variance $\sigma^2_h+\sigma^2_s$
so your integral should be $$\Phi\left(\frac{a -\mu_h-\mu_s}{\sqrt{ \sigma^2_h+\sigma^2_s}}\right)$$ where $\Phi(x)$ is the cumulative distribution function of a standard normal random variable