Problem 1

Let \(X_1, X_2,...,X_n\) 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 \(X_i\)’s. Find the distribution of \(Y\).

Solution 1:

Let \(\alpha\) be minimum value of \(X_i\). The probability of \(\alpha\) for any number is given by the total number of possibilities \(k^n\) minus the number of possibilities that a number greater than or less than \(\alpha\).

More generally:

  • Let the total number of posibilities be: \(k^n\)

  • Let the number greater than \(\alpha\) be: \((k - \alpha + 1)^n\)

  • Let the number of less than \(\alpha\) be: \((k - \alpha)\)

Therefore: \[ P(\alpha) = \frac{(k - \alpha + 1) (k-\alpha)}{k^n} \]

To simulate this:

minDist <- function(k,n) {
  Y = c()
  for (i in 1:n){
    X <- sample.int(n, k, replace = TRUE)
    Y[i] = min(X)
  }
  return(Y)
}

Var <- minDist(20, 100)
library(ggplot2)
qplot(Var, geom = 'histogram', bins = 100)

Problem 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.).

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..)

We use the pgeom function in R to calculate the proability that it will fail after 8 years given the 0.1 failure rate:

pgeom(7, 0.1, lower.tail = FALSE)
## [1] 0.4304672

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.

pexp(8, 0.1, lower.tail = FALSE)
## [1] 0.449329

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)

pbinom(0, 8, 0.1)
## [1] 0.4304672

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.

ppois(0, 8/10)
## [1] 0.449329