ERoman_HW8

Page 303

Exercise 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.)

u=1000
n=100
x= u/n

paste("The expected time for the first of these bulbs to burn out is: ",x)

## [1] "The expected time for the first of these bulbs to burn out is:  10"

Exercise 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) = (1/2)λe−λ|z|.

see hand written notes attached

Page 320 - 321 Exercise 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.

1. P(|X − 10| ≥ 2).
2. P(|X − 10| ≥ 5).
3. P(|X − 10| ≥ 9).
4. P(|X − 10| ≥ 20).

μ = 10 σ = (100/3)^(1/2)

1. P(|X − 10| ≥ 2). 2 = k*σ k = 2/σ

k = 2/((100/3)^(1/2))
1/k^2

## [1] 8.333333

2. P(|X − 10| ≥ 5). k = 5/σ

k = 5/((100/3)^(1/2))
1/k^2

## [1] 1.333333

3. P(|X − 10| ≥ 9). k = 9/σ

k = 9/((100/3)^(1/2))
1/k^2

## [1] 0.4115226

4. P(|X − 10| ≥ 20) k = 20/σ

k = 20/((100/3)^(1/2))
1/k^2

## [1] 0.08333333