# Thread: A problem from "First course in Probability"-Ross

1. ## A problem from "First course in Probability"-Ross

Given that X and Y are independent continuous positive random variables, express the density of the random variable Z in terms of the density of X and Y in each case.
a) Z = X/Y

b) Z = XY

c) Given that X~exp(lamda), Y~exp(u), evaluate the density obtained in part a)

2. Originally Posted by dingdong
Given that X and Y are independent continuous positive random variables, express the density of the random variable Z in terms of the density of X and Y in each case.
a) Z = X/Y

b) Z = XY

c) Given that X~exp(lamda), Y~exp(u), evaluate the density obtained in part a)
c) Something to get you started (and which should also give you some food for thought for a) and b)):

The cdf of Z is given by

$\displaystyle F(z) = \Pr(Z < z) = \Pr\left( \frac{X}{Y} < z \right) = \Pr(X < zY)$

$\displaystyle = \int_{y=0}^{y = + \infty} \int_{x=0}^{x = zy} \lambda \, \mu \, e^{-\lambda x} \, e^{-\mu y} \, dx \, dy$, $\displaystyle 0 \leq z < +\infty$.

The integrals should be routine at this level.

The pdf of Z is given by $\displaystyle f(z) = \frac{dF}{dz}$.