Calculating Conditional Beta Distribution with Binomial Parameters

[jimmy1]

I need to get the density function of a Beta distribution (call it B) with it's two parameters, X and Y, binomially distributed.

1) My first question is, would I be right in saying that the density function that I am looking for can be defined as a "conditional Beta distribution". ie. f(B|(X,Y))??
If this is right then how do I extend the usual conditional expectation formula, to that of conditional of two random variables.

2) My second question is to do with expectation and variance of the "conditional Beta distribution". If I know the mean of X and Y, could I just use these values to calculate the mean and variance of the "conditional Beta distribution". For example the mean of a Beta distribution is defined by a/(a+b), so if the mean of my binomial distributions, X and Y, were x1 and y1, then could I just say the the mean of my "conditional Beta distribution" would simply be x1/(x1 + y1)?
Similarly for the variance??

Any help would be great!

 

[EnumaElish]

1) E[beta|X,Y] is the first moment of F(β|X,Y) = Prob{beta < β|X,Y}.
2) As long as the beta density is nonlinear in X and Y, no.

 

[jimmy1]

I don't think I understand

E[beta|X,Y] is the first moment of F(β|X,Y) = Prob{beta < β|X,Y}.

I just need an expression for a random variable $$Z$$ which follows a Beta distribution, $$B(X,Y)$$ where $$X$$ and $$Y$$ follow Binomial distributions, so I'm looking for the distribuion of $$Z|X,Y$$.
If I had the situation $$Z|Y$$, then I could use the conditional expectation formula $$P(Z|Y)=P(Z,Y)/P(Y)$$, and then the distribution of $$Z$$ can be got as $$f_z(z)=\sum_{i=0}^nf(Z=z|y=i)f_y(y= i)$$

So how would I extend the formula $$f(Z|Y)=f(Z,Y)/f(Y)$$ to a similar formula for this $$f(Z|Y,X)= ??$$

 

[EnumaElish]

By replacing f(Z,Y) with f(Z,Y,X); replacing f(Y) with f(Y,X); and replacing "sum over y" with "sum over y, sum over x."