Conditional Probability: Converting CDF to PDF for Independent Random Variables

[joshthekid]

Basically I am wondering how you deal with a conditional cdf and turning that into a conditional pdf when the random variables are independent. I know that f(X|Y) =f(X)f(Y)/f(Y)=f(X)

I tried to derive this in a nice attached laTex document but it does not seem right to me.

Note(this is for a homework problem but this is only a derivation I am trying to use to solve it so I decided to post it here because it is not a textbook problem)

 

[RUber]

I have only seen this explained as
$f(X|Y) = \frac{f(X \cap Y)}{f(Y)}$ where $f(X \cap Y) = f(X)f(Y)$ by the definition of independence.
In your work, it seems like in part (6) you were taking the integral with respect to y, where you should be considering a fixed y and taking the integral with respect to x.
I have not put pen to paper, but it looks like that could get you something in a more recognizable form.

 

[Stephen Tashi]

The inequality g(x,y) < z can't necessarily be rewritten in the form x < h(y,z).

For example, the solution x^2 + y < z might require that x be in an interval of the form -a < x < a rather than in an interval of the form x < a.