If a random variable X has the moment generating function (1 - 2 t)^-2 for t 7.7

Question

If a random variable X has the moment generating function (1 - 2 t)^-2 for t 7.779). A student purchased a laptop computer at Joe's Discount Store. She also purchased the "Lucky 7" warranty plan that would replace the laptop at no cost if it needs 7 or more repairs in 3 years. Suppose the laptop requires repairs according to Poisson process with the average rate of one repair per 4 month. Find the probability that the laptop would not need to be replaced. That is, find the probability that the seventh time the laptop needs repair will be after 3 years, when the warranty expires. Find the probability that the seventh time the laptop needs repair will be during the second year of warranty.

Explanation / Answer

MGF = (1-2t)-2

= 1+2(2t)+3(4t2)+4(8t3)+...

Hence mean = 4

Var(x) = 12

P(X>7.779) = P(Z>3.779/rt 12)

=P(Z> 1.091)

= 0.5-0.3599

=0.1401

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No of repairs of a laptop is Poisson with mean 1 per 4 month

For 3 years average =9

a) Prob laptop is replaced = P(X>7 in 3 years)

= 0.6761

b) Prob 7th time is in 2nd year

= P(more than 6 in first year)+P(more than 5 in first year and 2 in second year)+P(4,3)+P(3,4)+P(2,5)...

i.e. If x is the no of repairs in first year and y in ii year

x+y =z

P(Z>=7) in average of 6

= 0.3937

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