Prob 11 Page 303 from “Intro to Probability Book”

Theorem tells us that If Xi ∼ exponential(λi), for i = 1, 2, . . . , n, and X1, X2, . . . , Xn are mutually independent random variables, then min{X1, X2, . . . , Xn} ∼ exponential (\(\sum_{x = 1}^{n} λi\))

E[Xi]=1/λi=1000, so λi=1/1000. Then minXi∼ Exp(λ), with λ=λ1+…+λ100=100/1000=1/10.

E[minXi] = 1/λ = 1/1/10 = 10.

ans:

Therefore, expected time for bulb to burn out = 10 hours

Prob 14 Page 303 from “Intro to Probability Book”

X1,X2 are independant random variables with exponential density and λ parameters

find Z = X - Y, in our case, we let Y = X2 and X = X1, see below:

Prob 1 Page 320-321 from “Intro to Probability Book”