Answer to Question #208307 in Statistics and Probability for Humphrey

Question

The time in hours a transistor lasts follows an exponential distribution with a mean of 100. Find the probability that a transistor lasts longer than

a) 15 hours.

b) 110 hours

c) 110 hours given that it lasts longer than 95 hours.

Answer 1

Let $X=$ the time a transistor lasts : $X\sim Exp(\lambda)$

Given $\lambda=\dfrac{1}{\mu}=\dfrac{1}{100}=0.01$

a)

$$P(X>15)=e^{-0.01(15)}\approx0.860708$$

b)

$$P(X>110)=e^{-0.01(110)}\approx0.332871$$

c)

$$P(X>110|X>95)=\dfrac{P((X>110)\cap (X>95))}{P(X>95)}$$

$$=\dfrac{P(X>110)}{P(X>95)}=\dfrac{e^{-0.01(110)}}{e^{-0.01(95)}}=e^{-0.01(15)}$$

$$=P(X>15)\approx0.860708$$