# Joint distribution of the max of the product of random vaiables

Suppose we have 4 positive independent random variables, $a_1, a_2, b_1$, and $b_2$. Moreover, $a_1, a_2$ are identically distributed and have a PDF denoted by $f_A(a)$. Similarly, $b_1, b_2$ have a PDF denoted by $f_B(b)$. I need to find the joint distribution of these two random variables:

Z=max($a_1 b_1, a_2 b_2$);

Y=$a_i$, where $i=argmax_{i \in{1,2}} a_i b_i$.

Any idea how can I approach this problem?

• Are you seeking the joint pdf of $Z$ and $Y$? – wolfies Oct 8 '16 at 7:21
• @wolfies Yes, the joint PDF of Y & Z. – Alammouri Oct 8 '16 at 18:51

## 1 Answer

Here is what I got so far, which is sufficient for my problem:

$F_{Z,Y}(z,y)=\mathbb{P} \left(Z<z,Y<y \right)=\mathbb{P} \left(a_1 b_1<z,a_2 b_2<z,a_1<y,a_2<y \right)$

$=\mathbb{P} \left( a_1< \min \left(\frac{z}{b_1},y\right) ,a_2< \min \left(\frac{z}{b_2},y\right) \right)$

Exploiting the i.i.d property:

$= \left( \int\limits_{0}^{\infty} F_{A}\left(\min \left(\frac{z}{b},y\right)\right)f_{B}(b) \ db\right)^2$