Let X and Y iid be ~ N(0,1)[Gaussian]

Question

Given Z=XY.Find the conditional pdf Z given X=x.

The way I calculated assumed W=X, then proceeded with Jacobian and replaced W=X in the final answer. Not sure whether it is right or not. Answer i got is $ \frac{1}{\sqrt{2\pi}}e^-{\frac{z^2}{2x^2}} $.

Could anyone help me if the procedure/answer is wrong

Answer 1

Not quite; your result doesn't integrate to $1$. For $x>0$,$$P(Z\le z|X\le x)=P\bigg(Y\le\frac{z}{x}\bigg)=\Phi\bigg(\frac{z}{x}\bigg),$$giving a pdf for $Z$ of $$\frac{1}{x}\phi\bigg(\frac{z}{x}\bigg)=\frac{1}{x\sqrt{2\pi}}\exp-\frac{z^2}{2x^2}.$$You can repeat this for $x<0$. In general, the PDF of $z$ is $$\frac{1}{|x|\sqrt{2\pi}}\exp-\frac{z^2}{2x^2}.$$