Let and be independent random variables representing the lifetime (in 100 hours)

Question

Let and be independent random variables representing the lifetime (in 100 hours) of Type A and Type B light bulbs, respectively. Both variables have exponential distributions, and the mean of X is 2 and the mean of Y is 3.

a)Find the probability that a Type A bulb lasts at least 300 hours and a Type B bulb lasts at least 400 hours.

b) Given that a Type B bulb fails at 300 hours, find the probability that a Type A bulb lasts longer than 300 hours.

c) What is the expected total lifetime of two Type A bulbs and one Type B bulb?

d) What is the variance of the total lifetime of two Type A bulbs and one Type B bulb?

Explanation / Answer

a) probability that a Type A bulb lasts at least 300 hours and a Type B bulb lasts at least 400 hours =

=P(X>300 and Y>400) =e-300/200*e-400/300 =0.0588

b)as X and Y are independent ; therfore P(X>3|Y=3) =P(X>3) =e-300/200 =0.2231

c)expected total lifetime of two Type A bulbs and one Type B bulb =2E(X)+E(Y)=2*2+3 =7 ( in 100 Hours)

d) variance of the total lifetime of two Type A bulbs and one Type B bulb =4*Var(X)+Var(Y)

=4*(2)2+(3)2 =25 ( in square hours)

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