Trivial exponential variables independence

Question

Let $X,Y,Z$ be exponentially distribute with parameter $\lambda_x, \lambda_y, \lambda_z$.

I tried to understand a proof which is using statement that events $\{A < min(B,C) \}$ and $\{B<C\}$ are independent. It assumes that it's a trivial fact that do not require justification but I'm sceptical about it. Could you please explain to me why does it holds ?

Answer 1

You appear to have changed your notation in midstream. Presumably, with $\ X,\ Y$ and $\ Z\ $ being the random variables, you would want to show that $\ \big\{X<\min(Y,Z)\big\}\ $ and $\ \big\{Y<Z\big\}\ $ are independent.

• Unless you impose further restrictions on $\ X,\ Y$ and $\ Z\ $, the result does not necessarily hold. I'm guessing that one of the hypotheses of the proof you're trying to understand is that $\ X,\ Y$ and $\ Z\ $ are independent random variables.
• If $\ X,\ Y$ and $\ Z\ $ are independent, then the result is true. I would certainly not call it "trivial", but the proof is sufficiently straightforward that its author may well have been justified in leaving his or her readers to verify it for themselves.

First note that $\ \big\{X<\min(Y,Z)\big\}= \big\{X<Y\big\}\cap\big\{X<Z\big\}\ $, so if $\ X,\ Y$ and $\ Z\ $ are independent \begin{align} P\big(X<\min(Y,Z)\big)&=P\big(\big\{X<Y\big\}\cap\big\{X<Z\big\} \big)\\ &=\int_0^\infty\int_x^\infty \int_x^\infty\lambda_x \lambda_y \lambda_ze^{-\lambda_xx} e^{-\lambda_yy} e^{-\lambda_zz}dzdydx\\ &=\lambda_x\int_0^\infty e^{-\left(\lambda_x+ \lambda_y+\lambda_z\right)x}dx\\ &=\frac{\lambda_x}{\lambda_x+ \lambda_y+\lambda_z}\ ,\\ P\big(Y<Z\big)&=\int_0^\infty\int_y^\infty\lambda_y \lambda_ze^{-\lambda_yy} e^{-\lambda_zz}dzdy\\ &= \frac{\lambda_y}{\lambda_y+\lambda_z}\ , \end{align} and \begin{align} P\big(\big\{X<\min(Y,Z)\big\}&\cap\big\{Y<Z\big\}\big)\\ &=P\big(\big\{X<Y\big\}\cap\big\{Y<Z\big\}\big)\\ &=\int_0^\infty\int_x^\infty\int_y^\infty \lambda_x \lambda_y \lambda_ze^{-\lambda_xx} e^{-\lambda_yy} e^{-\lambda_zz}dzdydx\\ &= \lambda_x \lambda_y\int_0^\infty e^{-\lambda_xx}\int_x^\infty e^{-\left(\lambda_y+\lambda_z\right)y}dydx\\ &=\frac{\lambda_x \lambda_y}{\lambda_y+\lambda_z}\int_0^\infty e^{-\left(\lambda_x+ \lambda_y+\lambda_z\right)x}dx\\ &= \frac{\lambda_x \lambda_y}{\left(\lambda_y+\lambda_z\right)\left(\lambda_x+ \lambda_y+\lambda_z\right)}\\ &=P\big(X<\min(Y,Z)\big) P\big(Y<Z\big)\ , \end{align} which shows that $\ \big\{X<\min(Y,Z)\big\}\ $ and $\ \big\{Y<Z\big\}\ $ are independent.