Question

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ = 1. What is a) the probability that a repair time exceeds 2 hours? b) the conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours?

Answer 1

Answer:

Step-by-step explanation:

Given

$\lambda =1$

Let X denote the time to repair a machine

(a)Probability that a repair time exceeds 2 hours

$P(X>2)=1-P(x\leq 2)$

$P(X>2)=1-\int_{0}^{2}\frac{1}{2}e^{-\frac{x}{2}}dx$

$=1+\left [ e^{-\frac{x}{2}}\right ]_0^2$

$P(X>2)=e^{-1}$

(b)Probability that a repair takes at least 10 hr, given that its duration exceeds 9 hours

$P\left ( X\geq 10|X>9\right )=\frac{P\left ( X\geq 10\right )}{P\left ( X>9\right )}$

$=\frac{\int_{10}^{\infty }\frac{1}{2}e^{-\frac{x}{2}}dx}{\int_{9}^{\infty }\frac{1}{2}e^{-\frac{x}{2}}dx}$

$=\frac{2e^{-\frac{1}{2}}|_{10}^{\infty }}{2e^{-\frac{1}{2}}|_{9}^{\infty }}$

$=e^{-\frac{1}{2}}$