Suppose the lifetime of a mobile smartphone processor can be modeled as an expon

Question

Suppose the lifetime of a mobile smartphone processor can be modeled as an exponential random variable with rate lambda = 0.1 broken processors / year (i.e. 1 out of 10 processors break per year).

a. What is the probability that a processor will die within two years?

b. What is the probability that it will survive for more than four years?

c. Assuming that the processor survives for at least one year, what is the probability that it survives at least two years?

d. What is the probability that exactly 15 broken processors will happen within a month? (Note: you can assume a typical month has 30 days and a typical year has 365 days).

e. What is the probability of having at most 3 broken processors within a week?

f. What is the mean number of broken processors within a year if the probability of exactly three broken processors within a year is 3%?

Explanation / Answer

a) CDF = 1-e-x

P(X <= 2) = 1-e-0.1*2 = 0.1813

b) P(X >=4) = 1 - P(X <=4) = 1 - (1 - e-0.1*4) = 0.6703

c) Using memoryless property,
P(X >= 2) = 1 - P(X<=2) = 1-0.1813 = 0.8187

d) This is based on poisson distribution
for a month, = 0.1/12
pmf = e-*(k/k!)
P(X = 15) = e-0.1/12*((0.1/12)15/15!) = 4.922 * 10-44

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