# Proper way to combine conditional probability distributions of the same random variable conditioned on a discrete variable ? (based on assumptions)

My question is the following. Let's say I have two probability distributions:

$f(x|b), g(x|c)$

$b$ and $c$ are discrete events while $x$ is a continuous variable, i.e., when the button b is pressed there is some distribution for the amount of rain fall the next day, $x$.

When the button $c$ is pressed there is a different distribution of rain fall the next day, $x$. Are there any strategies for estimating the distribution of rain fall if both buttons are pressed, i.e.,

$h(x|b,c)$ ?

And, what assumptions do those strategies rest on?

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You call both of your random variables (i.e., your events) x, but it might make more sense to call one y. Then we can ask, are x and y independent? If not, do you know how x and y are related? Without knowing the answers, it is not possible to get the joint probability distribution. That is, it is impossible to answer your question. –  Joel W. May 30 '12 at 20:46
Can you say a and b are independant?i.e. p(a)*p(b)=p(a and b) –  Seth May 30 '12 at 21:40
Seth - yes I can assume independence of b & c (I think you mean b and c as referred to in the question). –  BrainPermafrost May 30 '12 at 21:50
Joel W. - well, the reason I only use x is that they are the same random variable. –  BrainPermafrost May 30 '12 at 21:52
I don't see how this question makes sense. All you know is that b and c give different distributions for x. You don't even know what those distributions are. It could be that if b occurs you always get f(x|b) even when c occurs. Or what if c dominates then even if b occurs you get g(x|c). Those would be dependent cases. What would independence of b and c tell you? –  Michael Chernick May 31 '12 at 1:02
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