Introduction

The requirement for this assignment is to complete problems #11 and #14 on page 303 of probability text #1 on page 320-321.

Problem 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?

According to the “Distribution of the minimum of exponential random variables” section in https://en.wikipedia.org/wiki/Exponential_distribution:

The minimum of the minimum of the lambdas (1/the mean) is exponentially distributed and that lambda is the sum of all of the independent exponentially distributed lambdas.

lambda_sum <- 100*(1/1000)
expected <- 1/lambda_sum
print(paste0("The expected time to first fail is: ", expected, " hours"))
## [1] "The expected time to first fail is: 10 hours"

Problem 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

\[ f_Z(z)=(1/2)λe^{−λ|z|} \]

According to section 7.2 (pg. 291) of the textbook.

“Let X and Y be two continuous random variables with density functions f(x) and g(y), respectively. Assume that both f(x) and g(y) are defined for all real numbers. Then the convolution f ∗ g of f and g is the function given by”

$$

(f * g)(z) = _{-}^{+} f(z-y)g(y)dy

$$

Plugging in the exponential distribution and integrating results in the equality.

Problem 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.

\[ (a)P(|X−10|≥2) \\ (b)P(|X−10|≥5) \\ (c)P(|X−10|≥9) \\ (d)P(|X−10|≥20) \]

According to the book (pg. 306) Chebyshev’s Inequality states that:

$$

P(|X - u| ≥ ) = 2/2

$$ So going through each different epsilon.

sigma_sq <- 100/3

eps <- 2
print(paste0("Part A: ", round(sigma_sq/(eps^2), 4)))
## [1] "Part A: 8.3333"
eps <- 5
print(paste0("Part B: ", round(sigma_sq/(eps^2), 4)))
## [1] "Part B: 1.3333"
eps <- 9
print(paste0("Part C: ", round(sigma_sq/(eps^2), 4)))
## [1] "Part C: 0.4115"
eps <- 20
print(paste0("Part D: ", round(sigma_sq/(eps^2), 4)))
## [1] "Part D: 0.0833"