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CDF of a function of 2 random variables

  1. Nov 7, 2011 #1
    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.
     
  2. jcsd
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