A series of jointly distributed discrete random variables questions 
Suppose X and Y are jointly distributed discrete random variables, with probability mass function $P_x,y$(x,y)= ${k(1-\theta)^x(\theta)^y\over y! }$ where x,y = 0,1,2,... and $0<\theta<1$
1) Find out k 
  
  2) Are X and Y independent? why?
  
  3) Find $E(X^y)$

I am kinda stuck at first one, does this kind of probability mass function have anything to do with $\sum\sum f(x,y)=1$  ?
and second one is to test if it's p(x)p(y)?
thanks to any kind of help!

Yes, you need to find the "double sum," set it equal to $1$. First let us sum with respect to $y$.  We get
$$k(1-\theta)^x\sum_0^\infty \frac{\theta^y}{y!}.$$
You should recognize the inner sum as $e^\theta$. It is more familiar with different letters: the power series expansion of $e^x$ is $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$.
Now we want to sum with respect to $x$. So we want 
$$ke^\theta\sum_0^\infty(1-\theta)^x.$$
You will recognize the sum as an infinite geometric series with sum $\frac{1}{1-(1-\theta)}=\frac{1}{\theta}$.
For the second problem, yes the key thing, as you observed, is that the joint distribution function factors as $f(x)g(y)$. 
You did not ask about the third problem, so I will only say that it involves sum procedures closely related to the procedure used in the first problem.
