Suppose that the operating lifetime of a certain type of battery is an exponential random variable with parameter $\theta=2$ $($measured in years$)$. Find the probability that a battery of this type will have an operating lifetime of over 4 years.
My attempt:
We use $P(X\geq a+b|X\geq a)=P(X\geq a)$ $=$ $e^{-a/\theta}$
$f(t)=(1/2)e^{-t/2}$ for $t>0$ and $f(t)=0$
$P(X\geq 4|X\geq 2)=P(X\geq2+2|X\geq2)=P(X\geq2)$
$P(X\geq2)=(1/2)\int_{2}^\infty e^{-t/2}dt$
$=e^{-4/2}=e^{-2}=.13533$
I am not sure if my working is done correctly. Can someone please show me the correct working? If possible, can you show it the way I did it?
Best Answer
$f(t)=(1/2)e^{-t/2}$ for $t>0$
$$P(T>4) = \int_4^{\infty} f(t) dt$$
$$P(T>4) =1- \int_0^{4} (1/2)e^{-t/2} dt$$
$$P(T>4) =1+e^{-t/2}|4,0 $$ $$P(T>4) = 1+e^{-\frac{4}{2}}-e^{0} =1+e^{-2} -1 = e^{-2} = 0.135335283$$