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Density Function??

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  • Density Function??

    Can't figure out this one...

    A device containing 2 components fails when, and only when, both components fail. The lifetimes T1 and T2, of these components are independent with common density funciton f(t)=e^-t, t>0. The cost, X, of operating the device until failure is 2T1 + T2. What is the density function of X for x>0?

    Answer is e^(-x/2) - e^-x

    HELP PLEASE!!!!!!

  • #2
    I use convolution to solve this one. Rewrite the problem as Z = 2X + Y. U = 2X and V = Y. X = 1/2 * U, which gives us f(u) = 0.5*exp(-0.5*u). f(z) = int(0,z)(f(u)*f(z - u))du = int(0,z)(0.5*exp(-0.5*u)*exp(u)*exp(-z))du = 0.5*exp(-z)*int(0,z)(exp(0.5*u))du = exp(-z)*(exp(0.5*z) - 1) = exp(-0.5*z) - exp(-z).

    Switch the z back to x and you have your solution.

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    • #3
      Thanks for the reply, convolution makes sense. I saw a solution that used double integration aside from convolution. Can anyone post this alternative solution?

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