Investigation 11: More Battery Lifetimes (due Monday, October 25)

You may work with one other person on this assignment, handing in one report with both names. Word-processed reports are preferred to hand-written ones.

Suppose that the lifetime of a battery (in thousands of hours) is a random variable Y whose pdf is given by g(y)=2exp(-2y) for y>0, g(y)=0 for y<0.

a) Draw a sketch of this pdf.

b) Determine and sketch the cdf of this random variable.

c) Determine the expected (mean) value of this random variable.

d) Determine the median of this random variable.

e) Calculate the probability that this battery lasts for more than its mean value.

f) Calculate the probability that this battery lasts for more than 1 thousand hours.

g) Calculate the conditional probability that this battery lasts for more than 2 thousand hours, given that it lasts for more than 1 thousand hours.

h) Calculate conditional probability that this battery lasts for more than 3 thousand hours, given that it lasts for more than 2 thousand hours.

i) Generalize the previous two questions by calculating the probability that the battery lasts for more than (k+c) hours, given that it has already lasted for more than k hours, where c and k are positive real numbers.

j) How does your answer to i) compare to the (unconditional) probability that the batter lasts for c hours in the first place? (This is called the memoryless property.)