Question

if x1 and x2 are independent exponential random variables with respective parameters λ1 and λ2, find the distribution of Z = X1/X2 and (b) compute P(X1

Answer 1

$F_Z(z)=\mathbb P(Z\le z)$
$F_Z(z)=\mathbb P\left(\dfrac{X_1}{X_2}\le z\right)$
$F_Z(z)=\mathbb P(X_1\le X_2z)$
$F_Z(z)=\displaystyle\int_0^\infty\int_{x_1/z}^\infty f_{X_2}(x_2)f_{X_1}(x_1)\,\mathrm dx_2\,\mathrm dx_1$
$F_Z(z)=\displaystyle\int_0^\infty \lambda_1e^{-\lambda_1x_1}\left(\int_{x_1/z}^\infty\lambda_2e^{-\lambda_2x_2}\,\mathrm dx_2\right)\,\mathrm dx_1$
$F_Z(z)=\lambda_1\displaystyle\int_0^\infty e^{-\left(\lambda_1+\frac{\lambda_2}z\right)x_1\right)}\,\mathrm dx_1$
$F_Z(z)=\dfrac{\lambda_1}{\lambda_1+\frac{\lambda_2}z}$
$F_Z(z)=\dfrac{\lambda_1z}{\lambda_1z+\lambda_2}$