Elina Azrilyan

October 15th, 2019

11 and 14 on page 303 of probability text

  1. A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? (See Exercise 10.)

Solution: According to Exercise 10: - Let M be the minimum value of the Xj. - The density for the minimum value of the independent random variables is exponential with mean μ/n.

μ = 1,000 n = 100

Mean = 1000/100 = 10 hours

  1. Assume that X1 and X2 are independent random variables, each having an exponential density with parameter λ. Show that Z = X1 − X2 has density fZ(z)=(1/2)λe−^(λ|z|).