If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y?

Question

If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y ?

score 3 · Answer 1 · answered Apr 10 '15 at 06:17

3

I actually ran into this problem not so long ago. Replace the variables $X_1$ with X and $X_2$ with Y and you should receive the same answer.

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answered Apr 10 '15 at 06:17

Shrey

770

score 1 · Answer 2 · answered Apr 10 '15 at 06:16

Hint: First find the domain of $Z$. Then for any $z\in D_Z$ we have \begin{equation} P(Z\le z) = P(X/Y\le z) = P(X\le zY) = \int_{y\in D_y}P(X\le zy)f_Y(y)dy. \end{equation} Solve this and differentiate with respect to $z$ to find the density function of $Z$.

If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y?

2 Answers2