Multiplying pdf's: Statistics If I have two different independent continuous variables, X and Y. Is it legitimate to multiply their pdf's to form a pdf for the ordered pairs (X,Y) as f(x,y).  
I've plotted the surface and the volume under it is 1, and the key points (peaks and troughs seem to be in the correct position).
Is this a legitimate process?  What issues do I need to be aware of?  
 A: Multiplying the pdfs as you have done creates the joint pdf of the random variables $X$ and $Y$ or the pdf of the bivariate random variable $(X,Y)$ 
under the assumption that $X$ and $Y$ are independent random variables.
Whether $X$ and $Y$ are indeed independent or not is something that you
need to figure out from information that you have not provided
to us. Knowledge of just the individual distributions of $X$ and $Y$ does
not provide any information about independence.
As an example, suppose that $X$ and $Y$ are Bernoulli random variables
with parameter $\frac{1}{2}$. Your multiplication procedure gives a
joint distribution in which $(X,Y)$ takes on the $4$
values $(0,0), (0,1), (1,0), (1,1)$
with equal probability $\frac{1}{4}$.  However,  other joint 
distributions could also give you the same distribution for $X$ and $Y$,
and in these other cases $X$ and $Y$ are not independent.


*

*If $(X,Y)$ takes on values $(0,0)$ and $(1,1)$ with equal probability
$\frac{1}{8}$ and values $(0,1)$ and $(1,0)$ with equal probability $\frac{3}{8}$,
then $X$ and $Y$ are Bernoulli random variables
with parameter $\frac{1}{2}$.

*If $(X,Y)$ takes on values $(0,0)$ and $(1,1)$ with equal probability
$\frac{1}{2}$, then $X$ and $Y$ are Bernoulli random variables
with parameter $\frac{1}{2}$.  Note that $X = Y$.

*If $(X,Y)$ takes on values $(0,1)$ and $(1,0)$ with equal probability
$\frac{1}{2}$, then $X$ and $Y$ are Bernoulli random variables
with parameter $\frac{1}{2}$.  Note that $X = 1-Y$.
In summary, the multiplication is valid if $X$ and $Y$ are
independent random variables, but you need to provide some
justification as to why you believe that $X$ and $Y$ are
independent.
