2

I have a function like the following,

$p\left( y \right) = \int\limits_x {\int\limits_z {(Q({x^2} + y) + yz + z)dxdz} } $

Where, $Q(x) = \frac{1}{{2\pi }}\int\limits_x^\infty {{e^{ - \frac{{{t^2}}}{2}}}dt} $ and $x,y,z \in R$. I like to find $p(y|x)$ and $p(y|x = 0)$?

For, $p(y|x = 0)$ I put $x=0$. But I think I am wrong.

$p(y|x = 0) = \int\limits_z {(Q(y) + yz + z)dz} $

Samantha
  • 303
  • What have you tried? – David G. Stork Jul 23 '20 at 20:41
  • I edited the question description what I tried but I think this is not the correct way of doing. – Samantha Jul 23 '20 at 20:46
  • 1
    Just to be sure, is $Q(x^2 + y) + yz + z$ the conditional distribution of Y given $(X,Z)$? (That is what it seems to me). If so, then just integrate that wrt $z$ to obtain distribution of $Y|X$ and then plug in $x = 0$ in that to obtain the distribution for $Y| X = 0$. (In short, you seem to be on the right track) – sudeep5221 Jul 23 '20 at 22:11
  • Your definition of $Q(x)$ is a function of $t$, not $x$. Check the lower bound of the integral. $$\int_{\color{red}{t}}^\infty (\mathrm e^{-t^2/2}/{2\pi})~\mathrm d t=\operatorname{erfc}(\color{red}{t}/\sqrt{2\pi}) /(2\sqrt{2\pi})$$ – Graham Kemp Jul 23 '20 at 23:40

0 Answers0