Assume that the lifetime in hours of a particular car fan ca

Assume that the lifetime (in hours) of a particular car fan can be modeled with an exponential distribution with parameter 0.0003. Find the proportion of fans that will last for at least 10,000 hours.

Solution

Let X be the random variable that life time of a particular car fan.

Given that X ~ exp (paramteter = 0.0003)

Proportion of fans that will last for at least 10000 hours.

P(fans will last for at least 10000) = P(X >=10000)

.The p.d.f. of X is,

f(x) = *e^-x x >=0,

P(X >=10000) =   f(x) dx

=   *e^-x dx (x is from 1000 to )

=   e^-x dx    (x is from 1000 to )

= [ e^-x / - ]    (x is from 1000 to )

= - [ e^-x ]    (x is from 1000 to )

= - [ e^-  - e^{-0.0003*10000} ]

= - [ 0 - 0.049787 ]

P(X > = 10000) = 0.049787