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.

 

[sinat]

first time uses

 

[RPinPA]

This is a special case of what is called "order statistics", the lowest, 2nd lowest, 3rd lowest, of a set of random variables. You can google on that keyword to learn more about how they're calculated.

$T \leq x$ means either T1 is the smallest and is $\leq x$, or T2 is the smallest and is $\leq x$.

So $P(T \leq x) = P[ (T1 \leq x \text{ and } T2 <= T1) \text{ OR } (T2 \leq x \text{ and } T1 \leq T2) ]$