DATA605 WK7 Assignment 7

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: For \(1 ≤ j ≤ k\), \(m(j) = \frac{(k−j+1)^n−(k−j)}{k^n}\)

In other words

As Y is the minimum value of \(X_i\)’s over all of the \(X_i\)’s then the distribution function would be \(m(j) = P(Y = j)\).

We will need to count the number of ways that we can assign the \(X_i\)’s to values between \(j\) and \(k\) with at least one \(X_i\) being assigned to \(j\) divided by the total number of possible ways to assign \(X_i\)’s to values between \(1\) and \(k\).

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..)

\(P(X=n) = q^{n-1}p\) where \(n = 8\) and \(p = 0.1\)

# The probabilty of failure = 1/10
p = 0.1

# n trials
n = 8


dgeom(x = n, prob = p)

## [1] 0.04304672

\(E(X) = \frac{1}{p}\)

1 / p

## [1] 10

\(D(X) = \frac{1-p}{p^2}\)

sqrt((1 - p) / p**2)

## [1] 9.486833

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.

\(P(X \ge n) = e^{\frac{-k}{\mu}}\) where \(\lambda = 0.1\) and \(\mu = \frac{1}{\lambda} = 10\)

\(P(X \ge 8) = e^{\frac{-8}{10}}\)

pexp(8, rate = 0.1, lower.tail = F)

## [1] 0.449329

\(E(X) = \frac{1}{\lambda}\)

1 / 0.1

## [1] 10

\(D(X)\)

sqrt(1/0.1^2)

## [1] 10

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)

$b(n, p, k) =p^kq{n-k} $

\(P(K = 0) = b(8, 0.1, 0)\)

pbinom(0, 8, 0.1)

## [1] 0.4304672

\(E(X) = np\)

n = 8
p = 0.1

n*p

## [1] 0.8

\(D(X) = \sqrt{npq}\)

n = 8
p = 0.1
q = 1- p
sqrt(n * p * q)

## [1] 0.8485281

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.

\(P(X = k) \approx \frac{\lambda k}{k!}e^{−\lambda}\) where lambda \(8 / 10 = 0.8\)

Failing after 8 years, that is 8 or more years – \(P(X \ge 8)\)

\(\lambda\) is the expected number of successes in 8 years which is 0.

ppois(0, lambda=0.8, lower.tail = F)

## [1] 0.550671

\(E(X) = \lambda = 0.8\)

\(D(X) = sqrt(\lambda)\)

sqrt(0.8)

## [1] 0.8944272

Helpful Links:

Poisson Distribution