Homework SourseThe useful life X of a particular kind of machine is a rando
The useful life X of a particular kind of machine is a random variable with density function:
f(x) { (1.5)/(x^2)+1/9 if 3<=x<=9 0 if otherwise}
where 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 5 years.
B) Find the probability that a randomly selected machine will be useful for less than 7 years.
C) What is the expected useful life of the machine?
f(x) { (1.5)/(x^2)+1/9 if 3<=x<=9 0 if otherwise}
where 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 5 years.
B) Find the probability that a randomly selected machine will be useful for less than 7 years.
C) What is the expected useful life of the machine?
Solution
given pdf is f(x)=1.5/x2+1/9 3<x<9
a) p(x>5)= integ f(x) dx 5<x<9
=integ 1.5/x2+1/9 dx 5<x<9
=-1.5/x+1/9x 5<x<9
=4/9-1.5(1/9-1/5)
=4/9+2/15
=0.577777778
b)p(x<7) =?
integ f(x) dx 3<x<7
=1
c)
E(x)=?
integ xf(x) dx 3<x<7
=integ 1.5/x+(1/9)x dx 3<x<7
=1.5lanx+x2/18 3<x<7
=1.5lan(7/3)+40/9=5.71539
