CDF Query: Conditional CDF of S

[nikozm]

Hello.

I was wondering if the following is correct.

Let S= a*X/(b*X+c), where a,b,c are positive constants and X is a positive random variable. Also let H= h, where h is also a positive random variable (S and H are mutually independent).

Then, let F_{Z}(.) and f_{Z}(.) denote the CDF and PDF of Z, respectively. Thus, does the following holds true?

F_{S}(y) = ∫ F_{S|H}(a*X/(b*X+c) <= y | H) * f_{H}(h) dh.

Thank you in advance.

 

[mathman]

You say H and S are independent. In that case F_{S|H} = F{S}. Therefore the expression hold trivially.