Question

A water heater is designed to last an average of 16 years,the lifetime of such a heater can be modelled by an exponential random variables X with lambda value of 0.0625, suppose a heater is selected at random ,,,

What is the probability the heater will last at least 5 years

Suppose the heater last for 5 years what is the probability it will last for the other 5 years

Answer 1

Let $X=$ the lifetime of a heater: $X\sim Exp(\lambda).$

Given $\lambda=1/16=0.0625$

i.

$$P(X\geq 5)=e^{-\lambda(5)}=e^{-0.0625(5)}\approx0.731616$$

ii.

The memoryless property says that

$$P(X>r+t|X>r)=P(X>t)$$

$$P(X>5+5|X>5)=P(X>5)$$

$$=e^{-0.0625(5)}\approx0.731616$$