11 pg 303

A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out?

lambda<-1/1000 
n<-100
expectedmin<-1/(n*lambda)
expectedmin
## [1] 10

14 pg 303

Assume that X1 and X2 are independent random variables, each having an exponential density with parameter lambda. Show that Z = X1 − X2 has density fZ(z) = (1/2)lambda*e^−lambda|z|

\(f_Z(z)=\int_{-\infty}^{\infty} f_X1(z-x2) f_X2(x2)\; dx2\)

\(=\int_{0}^{\infty} \lambda^2 e^{-\lambda z}\; dx = \frac12\lambda e^{-\lambda z}\)

\(=\int_{-z}^{\infty} \lambda^2 e^{\lambda z}\; dx = \frac12\lambda e^{\lambda z}\)

1 pg 320 Let X be a continuous random variable with mean mu = 10 and variance sigma^2 = 100/3. Using Chebyshev’s Inequality, find an upper bound for the following probabilities. (a) P(|X − 10| >= 2)

k <- (2/sqrt(100/3))
1/k^2
## [1] 8.333333
  1. P(|X − 10| >= 5)
k <- (5/sqrt(100/3))
1/k^2
## [1] 1.333333
  1. P(|X − 10| >= 9)
k <- (9/sqrt(100/3))
1/k^2
## [1] 0.4115226
  1. P(|X − 10| >= 20)
k <- (20/sqrt(100/3))
1/k^2
## [1] 0.08333333