Conditional distribution of X given Y calculation HELP , PLEASE

Hi y'all - I don't suppose anybody could please help? Thanks a bunch

Could you post an image of the full question, including the options. Your text is hard to read.
You also say
f(x|y) = f(y|x) * f(x)
You forgot to divide the rhs by f(y)

Hi there,

Please find attached. Any help would be majorly appreciated , thank you !

Been a bit since I've done this, but since $X \sim M(1)$ then $f_X(x) = \displaystyle \int_{\mathbb{R}} f_{X,Y}(x,y) .dy = e^{-x}$.

And since $Y | X \sim M(x)$ then we have $f_{X,Y}(y|x) = \dfrac{f_{X,Y(x,y)}}{f_X(x)} = xe^{-xy}$

We can use this information to determine what $f_{X,Y}(x,y)$ must be.

Indeed, we can then use this to determine what $f_Y(y)$ must be by integrating marginally along $x \in (0, \infty)$.

Hence we seek what $f_{X,Y}(x|y) = \dfrac{f_{X,Y}(x,y)}{f_Y(y)}$ is.

Been a bit since I've done this, but since $X \sim M(1)$ then $f_X(x) = \displaystyle \int_{\mathbb{R}} f_{X,Y}(x,y) .dy = e^{-x}$.

And since $Y | X \sim M(x)$ then we have $f_{X,Y}(y|x) = \dfrac{f_{X,Y}(x,y)}{f_X(x)} = xe^{-xy}$

We can use this information to determine what $f_{X,Y}(x,y)$ must be.

Indeed, we can then use this to determine what $f_Y(y)$ must be by integrating marginally along $x \in (0, \infty)$.

Hence we seek what $f_{X,Y}(x|y) = \dfrac{f_{X,Y}(x,y)}{f_Y(y)}$ is.

Ah , I see what you mean now after your edit... How do we go about solving f(y) do you know? I know you mention integrating, but what do we integrate?

I think I know that f(x,y) is xe^-3x but from there I am not sure where to go?

Don't think I agree with this. Show your working out?

Yes, I think it should actually be xe^-2xy after we have multipled them together. So I think that after integration this becomes e^-y(xe^x - e^x)

Any thoughts on where we go from there ? In order to establish the distribution?

Yes, I think it should actually be xe^-2xy after we have multipled them together.

Hi all resolved now. The answer is xe^(-x(y+1)) as suggested by mqb and this is a gamma (2, y+1) distribution which is equal to f( x given y) . Thanks all !!

Agreed.