During my exam there was the following question which I could not answer:

Let $X_1, X_2$ be real valued random variables. Assume that $X_1$ is exponentially distributed. Given that $\{X_1=a\}$, $X_2$ is normally $N(a,a^2)$ distributed.

How are $(X_1,X_2)$ and $(X_1,X_2)$ distributed? How is $X_2$ distributed?

I thought that maybe the theorem of total probability would be useful for the second question but I tried to no avail.

I appreciate all help.

  • $\begingroup$ The first question asks two times the same thing, i.e. the distribution of $(X_1,X_2)$. Is there a typo? $\endgroup$ – Jimmy R. Apr 26 '16 at 17:09
  • $\begingroup$ Hey Jimmy, no this indeed the proper formulation of the question. Our professor likes to add trick questions to see if the students have understood the subject. $\endgroup$ – Theodor Johnson Apr 26 '16 at 17:11
  • $\begingroup$ Hmm, I do not see the trick. Assuming exponential(λ) then is this what you tried: $f_{X_2}(x_2)=\int_{\Bbb R}f_{X_2\mid X_1}(x_2\mid x_1=a)f_{X_1}(a)da=\int_{0}^{+\infty}\frac{1}{a\sqrt{2π}}e^{-\frac1{2a^2}(x_2-a)^2}λe^{-λa}\;da$ $\endgroup$ – Jimmy R. Apr 26 '16 at 17:14

The joint pdf of $(X_1, X_2)$ is : $f(x,y) = f_{X_1}(x) f_{X_2}(y \space | X_1=x)$

and you know the RHS by assumption.

Once you know the joint pdf you can derive the pdf of $X_2.$


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