# Thread: Probability density function problem

1. ## Probability density function problem

Here is a question that I encounter today.

two continuous random variables A and B whose joint pdf is given by
f(a,b)=b/(e^(a+b)), a>0, b>0.

Question
1.find pdf of U=2-B^2
2.find the pdf of W=min(A,B)

Wonder could anybody help?

2. $\displaystyle (1)$ Find the marginal $\displaystyle f_B,$ now do the transform.

$\displaystyle (2)$ $\displaystyle F_W(w) = 1-P(A>w,B>w),$ and $\displaystyle \frac{d}{dx}F_W = f_W.$

3. The rvs A and B are independent by inspection and their marginals are

$\displaystyle f_A(a)=e^{-a}I(a>0)$ which is an exponential

and

$\displaystyle f_B(b)=be^{-b}I(b>0)$ which is a Gamma

4. Originally Posted by Anonymous1
$\displaystyle (1)$ Find the marginal $\displaystyle f_B,$ now do the transform.

$\displaystyle (2)$ $\displaystyle F_W(w) = 1-P(A>w,B>w),$ and $\displaystyle \frac{d}{dw}F_W = f_W.$

and

$\displaystyle F_W(w) = 1-P(A>w,B>w)=1-F_A(w)F_B(w)$

and $\displaystyle \frac{d}{dw}F_W(w) = f_W(w).$