Statistics: Deriving a Joint Probability Function From a Definition of Other PDF's Here's a particular question I'm trying to understand from the lecture notes.
It says: 

Assume that $Y$ denotes the number of bacteria per cubic
  centimeter in a particular liquid and that $Y$ has a Poisson
  distribution with parameter $x$. Further assume that $x$ varies from
  location to location and has an exponential distribution with
  parameter $β = 1$. 
   -Find $f_{X,Y}(x,y)$, the joint probability function of $X$ and $Y$.

In the lecture slides, it says:
$$f_{X,Y}(x,y) = f_{Y\mid X}(y\mid x)f_X(x)$$
Where $f_{Y\mid X}(y\mid x)$ is the PDF for the Poisson distribution, and $f_X(x)$ is the PDF for the exponential with $\beta = 1$.
I'm not sure how $f_{X,Y}(x,y) = f_{Y\mid X}(y\mid x)f_X(x)$ came to be. Is there anything for me to look for in the question that would hint to using this form? 
Any help would be appreciated. 
Thanks.

Is it true that if we extended this to the case for a random vector (X,Y,Z) with a pdf $f_{X,Y,Z}(x,y,z)$, (also X, Y, Z have their own pdf's with the same notation as above) we would have the conditional expectation be equal to: $$f_{X,Y|Z}(x,y|z) = \frac{f_{X,Y,Z}(x,y,z)}{f_{Z}(z)}?$$
