DATA 605 - Homework 8

Page 303 Exercise 11

Solution

num_lightbulb <- 100
lifetime_lightbulb <- 1000

burnout <- lifetime_lightbulb/num_lightbulb

burnout

## [1] 10

Page 303 Exercise 14

Solution

\[Z = X_1 - X_2\]

\[X_2 = X_1 - Z\] $$$$

\[fZ(z) = \int_{\infty}^{\infty}fX_1(X_1)fX_2(X_1 - Z)dx_1\]

\[fZ(z) = \int_{Z}^{\infty}\lambda e^{-\lambda X_1}\lambda e^{-\lambda(X_1 - z)} dx_1\]

\[fZ(z) = \int_{Z}^{\infty}\lambda^2 e^{-2 \lambda X_1 + \lambda Z} dx_1\]

\[fZ(z) = \int_{Z}^{\infty} (1/2)\lambda e^{- \lambda Z}\]

Page 320-321 Exercise 1

#Solution

upper_bound_func <- function(x){
  result <- (100 / (3 * x**2))
  return (result)
}

prob_a <- upper_bound_func(2)
prob_b <- upper_bound_func(5)
prob_c <- upper_bound_func(9)
prob_d <- upper_bound_func(20)

prob_a

## [1] 8.333333

prob_b

## [1] 1.333333

prob_c

## [1] 0.4115226

prob_d

## [1] 0.08333333