DATA 605 FUNDAMENTALS OF COMPUTATIONAL MATHEMATICS

Assignment 8: Convolution & Chebyshev’s Inequality

Kyle Gilde

10/20/2017

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11 and #14 on page 303; 1 on page 320-321

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.)

N <- 1000
bulbs <- 100
N/bulbs
## [1] 10

14. Assume that X1 and X2 are independent random variables, each having an exponential density with parameter \(\lambda\). Show that Z = X1 - X2 has density

\[\begin{equation} fZ(z) = (1/2)\lambda e^{-\lambda|z|} \end{equation}\] \[\begin{equation} f_X(x) = f_Y(x) = \begin{cases}{\lambda e^{-\lambda x},} & \text{if } x\geq 0, \\ {0} & \text{otherwise;}\end{cases} \end{equation}\] \[\begin{equation} f_Z(z) = \int_{-\infty}^{+\infty} f_X(z + y) f_Y(y) \text{dy} \end{equation}\] \[\begin{equation} f_Z(z) = \int_{0}^{z} \lambda e^{-\lambda (z + y)} \lambda e^{-\lambda y} dy \end{equation}\] \[\begin{equation} f_Z(z) = \int_{0}^{z} \lambda^2 e^{-\lambda z} dy \end{equation}\] \[\begin{equation} f_Z(z) = \frac{\lambda}{2} e^{-\lambda |z|} \end{equation}\] \[\begin{equation} f_X(x) = f_Y(x) = \begin{cases}{(1/2)\lambda e^{-\lambda|z|},} & \text{if } z\geq 0, \\ {0} & \text{otherwise.}\end{cases} \end{equation}\]

8.2: 1. Let X be a continuous random variable with mean \(\mu\) = 10 and variance \(\sigma^2\) = 100/3. Using Chebyshev’s Inequality, find an upper bound for the following probabilities.

  1. P(|X - 10|>= 2).
VarX <- 100/3
(sigma <- sqrt(VarX))
## [1] 5.773503
epsilon <- 2
(k <- epsilon/sigma)
## [1] 0.3464102
(chebyshev <- pmin(1/k^2, 1))
## [1] 1
  1. P(|X - 10|>= 5).
VarX <- 100/3
(sigma <- sqrt(VarX))
## [1] 5.773503
epsilon <- 5
(k <- epsilon/sigma)
## [1] 0.8660254
(chebyshev <- pmin(1/k^2, 1))
## [1] 1
  1. P(|X - 10|>= 9).
VarX <- 100/3
(sigma <- sqrt(VarX))
## [1] 5.773503
epsilon <- 9
(k <- epsilon/sigma)
## [1] 1.558846

(chebyshev <- 1/k^2)
## [1] 0.4115226
  1. P(|X - 10|>= 20).
VarX <- 100/3
(sigma <- sqrt(VarX))
## [1] 5.773503
epsilon <- 20
(k <- epsilon/sigma)
## [1] 3.464102
(chebyshev <- 1/k^2)
## [1] 0.08333333