Problem 0.1
Suppose X and Y are two independent exponential random variables with respective densities given by(λ,θ>0) f(x) =λe^(−xλ) for x>0 and g(y) =θe^(−yθ) for y>0.
(a) Show that Pr(X<Y) =∫f(x){1−G(x)}dx {x=0, infinity] where G(x) is the cdf of Y, evaluated at x [that is,G(x) =P(Y≤x)].
(b) Using the result from part (a), show that P(X<Y) =λ/(θ+λ).
(c) You install two light bulbs at the same time, a 60 watt bulb and a 100 watt bulb. The lifetime of the 60 watt bulb has an exponential density with an average lifetime 1500 hours. The 100watt bulb also has an exponential density with average lifetime of 1000 hours. What is the probability that the 100 watt bulb will outlast the 60 watt bulb?
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