You sum on the discrete one and integrate on the continuous
I prefer to use the term density for only continuous RVs, but not everyone does that.
"Suppose X be a discrete random variable and Y be a continuous random varaible. Let f_X,Y (x,y) denote the joint density function of X and Y..."
I'm not sure if I understand the meaning of joint density here. In the case above, what does the joint density mean? What are the properties of it? Does it still double integrate to 1 (like for the jointly continuous case)?
From my probability class, I remember that a joint density function is only defined in the case when X and Y are BOTH continuous random variables. But when one random vaiable is continuous and the other discrete, how do we even define the joint density function?
Hopefully someone can explain this.
∫ ∑ f_X,Y (x,y) dy = 1
where the sum is over the support of X and the integral is over the support of Y?
Does it matter whether we do the sum first or the integral first? (i.e. can we interchange the ∫ and ∑ ?)