The useful life of a particular kind of machine is a random

The useful life of a particular kind of machine is a random variable with density function

f(x)= .8/x^2+1/6 if X is greater than or equal to 3 and less than or equal to 8, otherwise 0.

where X is the number of years a randomly selected machine stays in use.

A. find the probability that a randomly selected machine will be useful for more than four years.

B. find the probability that a randomly selected machine will be useful for less than 5 years.

C. what is the expected useful life of the machine?

Solution

A._4..8 .8/x^2+1/6 dx=-.8/x+1/6x|4..8=-.8/8+8/6+.8/4-4/6

B._3..5 .8/x^2+1/6 dx=-.8/x+1/6x|3..5=-.8/5+5/6+.8/3-3/6

C._3..8 (.8/x^2+1/6)x dx=_3..8 .8/x+x/6 dx=.8ln(x)+x²/12|3..8=.8ln(8/3)+64/12-9/12=.8kn(8/3)+55/12