(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

Exponential Dist

[itex]f(T) = \lambda e^{-\lambda T}[/itex]

[itex] F(T) = 1 - e^{-\lambda T}[/itex]

Rayleigh-dist

[itex]f(T) = \frac{T}{\alpha^2} e^{\frac{-T}{2\alpha^2}}[/itex]

[itex]F(T) = 1 - e^{\frac{-T}{2\alpha^2}}[/itex]

3. 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:

[itex] T = min(T_1,T_2)[/itex]

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

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# CDF of a function of 2 random variables

Can you offer guidance or do you also need help?

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