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? (See Exercise 10.)

For n independent random variables each with exponential density µ and M being the minimum value of the each random variable, the density for M is exponential with mean µ/n.

µ = 1000 n = 100

density(M) = 1000/100

The expected time for the first of the bulbs to run out is 10 hours.

(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

\(fZ(z) = \fracλ{2}e^{−λ|z|}\)

\(\int_{-inf}^\inf fX1(X1)*fX2(X1-Z)\)

\(\int_{Z}^\inf λe^{-λX1}λe^{-λ(X1-Z)}dX1\)

\(\int_{Z}^\inf λ^2e^{-λ(2X1-Z)}dX1\)

$ = e^{λ|z|}dX1$

1 Let X be a continuous random variable with mean µ = 10 and variance σ2 = 100/3. Using Chebyshev’s Inequality, find an upper bound for the following probabilities. (a) P(|X − 10| ≥ 2). (b) P(|X − 10| ≥ 5). (c) P(|X − 10| ≥ 9). (d) P(|X − 10| ≥ 20).

Chebyshev Inequality:

P(|X-µ| >= E) <= σ2/E2

var_sqrd <- 100/3

#a 
(var_sqrd)/(2^2)
## [1] 8.333333
#b
(var_sqrd)/(5^2)
## [1] 1.333333
#c
(var_sqrd)/(9^2)
## [1] 0.4115226
#d
(var_sqrd)/(20^2)
## [1] 0.08333333