11 and #14 on page 303 of probability text

  1. A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the ???rst of these bulbs to burn out? (See Exercise 10.)
n <-  1000
litbulbs <-  100

n/litbulbs
## [1] 10
  1. Assume that X1 and X2 are independent random variables, each having an exponential density with parameter ??. Show that Z = X1 ???X2 has density

\[\begin{equation} fZ(z) = (1/2)\lambda e^{-\lambda|z|} \end{equation}\]

\[\quad \int _{ 0 }^{ \infty }{ { \lambda }^{ 2 }{ e }^{ -\lambda z }dx } =\frac { 1 }{ 2 } \lambda { e }^{ -\lambda z }\]

\[\quad \int _{ -z }^{ \infty }{ { \lambda }^{ 2 }{ e }^{ \lambda z }dx }=\frac { 1 }{ 2 } \lambda { e }^{ \lambda z }\]

1 on page 320-321

1, Let X be a continuous random variable with mean µ = 10 and variance ??2 = 100/3. Using Chebyshev’s Inequality, ???nd an upper bound for the following probabilities.

  1. 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