Question

2. -30 a) The joint pdf of random variables X and Y is given by f(x,y) = 27ye-3 y<x<0, y >0. Show that the joint moment generating function(mgf) of X and Y is 27 M(t1, tz) = tı <3, tı + t, <3 (3 - tı) (3 - 7ı - t2) Use the joint mgf to obtain Cov(X,Y). b) Let X1, X2, X3 be independent random variables representing the lifetime of 3 electronic components with the following pdf, where X in measured in years, f(x) = 2xe-7, => 0. (i) Calculate the probability that the survival time of the first component to fail will be lesser than 0.3 year. (ii) Find the probability that the median lifetime of the 3 components will be between 0.4 to 1 year. (iii) The 3 components are used in a system and the system will function as long as at least 2 of the 3 components are functioning. Calculate the probability that the system will survive longer than 0.5 year.

Answer 1

Now solve the inner integral by letting u=y and dv=e^(t2y)dy, we have

then

so that

Now, if t1+t2<3 and t1<3 then all three integrals converge and we have

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