# CDF of a function of 2 random variables

vortmax

## Homework Statement

Two toys are started at the same time each with a different battery. The first battery has a lifetime that is exponentially distributed with mean 100 min; the second battery has a lifetime that is Rayleigh-distributed with a mean 100 minutes.

a) Find the CDF to the time T until the battery in a toy first runs out
b) Suppose that both toys are still operational at 100 minutes. Find the CDF of the time T2 that subsequently elapses until the battery in a toy first runs out
c) in part b, find the cdf to the total time that elapses until a battery first fails.

## Homework Equations

Exponential Dist

$f(T) = \lambda e^{-\lambda T}$
$F(T) = 1 - e^{-\lambda T}$

Rayleigh-dist

$f(T) = \frac{T}{\alpha^2} e^{\frac{-T}{2\alpha^2}}$
$F(T) = 1 - e^{\frac{-T}{2\alpha^2}}$

## The Attempt at a Solution

First, I calculated values for lambda and alpha based on the means ... but this part isn't entirely necessary to the final solution.

a) I need to find the cdf of the function:

$T = min(T_1,T_2)$

where $T_1$ and $T_2$ are the two RV's respectively... but I'm really at a loss about how to proceed.