Assignment 8

  1. 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.)

The expected value of the Exponential distribution is: λe^(-λx)

n = 100
rate = 1000
n_rate = rate / n

paste0("The first bulb will burn out at ", n_rate, " hours")
## [1] "The first bulb will burn out at 10 hours"
  1. 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|.

I’m not sure where to approach this problem

p320-321 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.(on page 320-321) P(|X−10|≥2) P(|X−10|≥5) P(|X−10|≥9) P(|X−10|≥20)

\[Chebyshev’s Inequality = P(|X-\mu | \le k\sigma) \le \frac{1}{k^2}\]

\(\mu = 10\) and \(\sigma = \sqrt{100/3}\)

upper_bound <- function(x) {
k <- x/sqrt(100/3)
u_b <- 1/k^2
paste0("upper bound is ", format(u_b,digits=4)) 
}

P(|X−10|≥2)

upper_bound(2)
## [1] "upper bound is 8.333"

P(|X−10|≥5)

upper_bound(5)
## [1] "upper bound is 1.333"

P(|X−10|≥9)

upper_bound(9)
## [1] "upper bound is 0.4115"

P(|X−10|≥20)

upper_bound(20)
## [1] "upper bound is 0.08333"