1. Let X1, X2, . . . , Xn be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the Xi’s. Find the distribution of Y.

To answer this question, we can plot a histogram and observe Y’s distribution.

# Define the variables we need to simulate the problem.
n <- 50000
k <- 50
samples <- 10
Y <- c()

# Simulate the problem.
for (count in 1:n) {
  Xn <- sample(1:k, samples, TRUE)
  Y <- c(Y, min(Xn))
}

# Plot the results as a histogram so we can observe the distribution of Y.
distribution_of_y <- hist(Y, col = '#A52A2A', main = 'Distribution of Y')

Answer: The distribution of Y is positively skewed.

2. Your organization owns a copier (future lawyers, etc.) or MRI (future doctors). This machine has a manufacturer’s expected lifetime of 10 years. This means that we expect one failure every ten years. (Include the probability statements and R Code for each part.).

  failure_probability <- 0.1

2. a. What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as a geometric. (Hint: the probability is equivalent to not failing during the first 8 years..)

Geometric Model

  # Calculate the probability that the machine will fail after 8 years using a geometric model.
  machine_failure_geo_probability <- pgeom(8, failure_probability, lower.tail = FALSE)
  machine_failure_geo_probability
## [1] 0.3874205
  # Calculate the standard deviation for the geometric model.
  geometric_standard_deviation <- sqrt((1 - failure_probability) / (failure_probability^2))
  geometric_standard_deviation
## [1] 9.486833

Geometric Probability Formula: \(P(X=n) = q^{n-1} * p\)

Geometric Standard Deviation: 9.486833

Answer: Using a geometric model, the probability that the machine will fail after 8 years is 0.3874205.

2. b. What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as an exponential.

Exponential Model

  # Calculate the probability that the machine will fail after 8 years using an exponential model.
  machine_failure_exp_probability <- pexp(8, failure_probability, lower.tail = FALSE)
  machine_failure_exp_probability
## [1] 0.449329
  # Calculate the standard deviation for the exponential model.
  exponential_standard_deviation <- sqrt(1 / failure_probability^2)
  exponential_standard_deviation
## [1] 10

Exponential Probability Formula: \(P(X \le k) = 1 - e^{\frac{-k}{\mu}}\), where \(\mu = \frac{1}{\lambda}\)

Exponential Standard Deviation: 10

Answer: Using an exponential model, the probability that the machine will fail after 8 years is 0.449329.

2. c. What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as a binomial. (Hint: 0 success in 8 years)

Binomial Model

  # Calculate the probability that the machine will fail after 8 years using an binomial model.
  machine_failure_binomial_probability <- pbinom(0, 8, failure_probability)
  machine_failure_binomial_probability
## [1] 0.4304672
  # Calculate the standard deviation for the binomial model.
  binomial_standard_deviation <- sqrt(8 * failure_probability * (1 - failure_probability))
  binomial_standard_deviation
## [1] 0.8485281

Binomial Probability Formula: \(P(X=k) = {n \choose k} p^{k}q^{n-k}\)

Binomial Standard Deviation: 0.8485281

Answer: Using a binomial model, the probability that the machine will fail after 8 years is 0.4304672.

2. d. What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as a Poisson.

Poisson Model

  # Calculate the probability that the machine will fail after 8 years using an Poisson model.
  machine_failure_poisson_probability <- ppois(0, 0.8)
  machine_failure_poisson_probability
## [1] 0.449329
  # Calculate the standard deviation for the Poisson model.
  poisson_standard_deviation <- sqrt(8 * failure_probability)
  poisson_standard_deviation
## [1] 0.8944272

Poisson Probability Formula: \(P(X=k) = \frac{\lambda^{k}}{k!}e^{-\lambda}\)

Poisson Standard Deviation: 0.8944272

Answer: Using a Poisson model, the probability that the machine will fail after 8 years is 0.449329.