11. 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?
1000/100
## [1] 10
14. Assume that X1 and X2 are independent random variables, each having an exponential density with parameter λ. Show that Z = X1 − X2 has density
\[\begin{equation} fZ(z) =\frac { 1 }{ 2 }\lambda e^{-\lambda|z|} \end{equation}\]
\[\quad \int _{ 0 }^{ \infty }{ { \lambda }^{ 2 }{ e }^{ -\lambda z }dx } =\frac { 1 }{ 2 } \lambda { e }^{ -\lambda z }\]
\[\quad \int _{ -z }^{ \infty }{ { \lambda }^{ 2 }{ e }^{ \lambda z }dx }=\frac { 1 }{ 2 } \lambda { e }^{ \lambda z }\]
1. Let X be a continuous random variable with mean µ = 10 and variance σ^2 = 100/3. Using Chebyshev’s Inequality, find an upper bound for the following probabilities.
(a) P(|X − 10| ≥ 2).
v<-sqrt(100/3)
n<-2/v
1/n^2
## [1] 8.333333
(b) P(|X − 10| ≥ 5).
n<-5/v
1/n^2
## [1] 1.333333
(c) P(|X − 10| ≥ 9).
n<-9/v
1/n^2
## [1] 0.4115226
(d) P(|X − 10| ≥ 20).
n<-20/v
1/n^2
## [1] 0.08333333