Thread: Independence rules

Hello,

It is a simple probability question but as it has been too long since I took a statistics course, I am not able to solve the problem. I will be appreciated if you can give an idea for the problem.

I have a joint probability distribution on the variables x and y given the parameters a and b:



x and y are assumed to be independent. and are multinomial distributions separately. Therefore, a is defined for x and b is defined for y. Therefore the equation can be written as given below (because of the independence between x and y):



Is it also possible to derive this equation as given below by discarding b for x, and a for y: ?



Because b does not mean anything for x, and a does not mean anything for x.

Thanks a lot in advance!

the short answer is yes. Your intuition that irrelevant parameters can be removed is correct.



More formally, for continuous variables (without loss of generality):



since you have said that p(x|a,b) is does not actually vary with b, it can go outside the integral:




but the integral of a pdf is always 1:



which is the result you wanted to show. You can do the same thing for p(y|b,a)

Very good proof. Thank you very much!