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).

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.)

lambda <- 100/1000
expected_lifetime <- 1/lambda
expected_lifetime
## [1] 10

14

4 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| .

lambda <- 1

f_x1 <- function(x) lambda * exp(-lambda * x)
f_x2 <- function(x) lambda * exp(-lambda * x)
f_z <- function(z) (1/2) * lambda * exp(-lambda * abs(z))

print(f_z)
## function(z) (1/2) * lambda * exp(-lambda * abs(z))
mu <- 10
sigma2 <- 100/3
sigma <- sqrt(sigma2)

k_a <- 2/sqrt(100/3)
k_b <- 5/sqrt(100/3)
k_c <- 9/sqrt(100/3)
k_d <- 20/sqrt(100/3)

bound_a <- ifelse(1/(k_a^2) > 1, 1, 1/(k_a^2))
bound_b <- ifelse(1/(k_b^2) > 1, 1, 1/(k_b^2))
bound_c <- ifelse(1/(k_c^2) > 1, 1, 1/(k_c^2))
bound_d <- ifelse(1/(k_d^2) > 1, 1, 1/(k_d^2))

bound_a
## [1] 1
bound_b
## [1] 1
bound_c
## [1] 0.4115226
bound_d
## [1] 0.08333333