Please refer to the Assignment 7 Document.

library(tidyr)
library(dplyr)
library(gtools)

1 Problem Set 1 (Sec5.1 Ex6)

Let \(X_{1}, X_{2}, \cdots , X_{n}\) 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 \(Y\).

1.1 PS1 Answer

Given that \(i \in [1,n], \;\forall X_{i} \in [1,k], \;Y=min(X_{i})\)

For \(Y=1,\) the probability for an \(X_{i}=1\) is \(\frac{1}{k}\) and the probability for an \(X_{i}>1\) is \(\frac{k-1}{k}\).

\[\therefore P(Y=1)=P(min (X_{i})=1)=1-(\frac{k-1}{k})^{n}\] \[=(\frac{k-0}{k})^{n}-(\frac{k-1}{k})^{n}\]

For \(Y=2,\) the probability for an \(X_{i}>2\) is \(\frac{k-2}{k}\).

\[\therefore P(Y=2)=P(min (X_{i})=2)=1-P(Y=1)-P(Y>2)\] \[=1-(1-(\frac{k-1}{k})^{n})-(\frac{k-2}{k})^{n}\] \[=(\frac{k-1}{k})^{n}-(\frac{k-2}{k})^{n}\]

For \(Y=3,\) the probability for an \(X_{i}>3\) is \(\frac{k-3}{k}\).

\[\therefore P(Y=3)=P(min (X_{i})=3)=1-P(Y=1)-P(Y=2)-P(Y>3)\] \[=1-(1-(\frac{k-1}{k})^{n})-((\frac{k-1}{k})^{n}-(\frac{k-2}{k})^{n})-(\frac{k-3}{k})^{n}\] \[=(\frac{k-2}{k})^{n}-(\frac{k-3}{k})^{n}\]

Therefore, by induction, the distribution of Y is \[P(Y=y)=P(min (X_{i})=y)=(\frac{k-y+1}{k})^{n}-(\frac{k-y}{k})^{n}\;\;\;for\,y \in[1,k]\,,\;k\in \mathbb{Z}^{+}\]

2 Problem Set 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.)

(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.

(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.)

(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.

2.1 PS2 Answer

lifetime = 10
p = 1/lifetime #probability of machine failure
q = 1-p
n = 8

2.1.1 a. Geometric

Assuming the unit is annual. Geometric pmf is \(p(1-p)^{n-1}\), \(E[X]=\frac{1}{p}\) and \(V[X]=\frac{1-p}{p^{2}}\).

The probability that the machine will fail after 8 years is \(0.4304672\).

The expected value is \(10\) and the standard deviation is \(9.486833\).

q^8 #probability of 0 machine failures in 8 years

## [1] 0.4304672

pgeom(n-1,p,lower.tail = FALSE)

## [1] 0.4304672

1/p #expected value

## [1] 10

sqrt((1-p)/(p^2)) #sd

## [1] 9.486833

2.1.2 b. Exponential

Exponential pdf is \(\lambda e^{-\lambda x}\) for \(x\geq0\) and 0 otherwise. \(E[X]=\frac{1}{\lambda}\) and \(V[X]=\frac{1}{\lambda^{2}}\), where \(\lambda=1/10=p\).

The probability that the machine will fail after 8 years is \(0.449329\).

The expected value is \(10\) and the standard deviation is \(10\).

pexp(n, p, lower.tail = FALSE)

## [1] 0.449329

1/p #E[X]

## [1] 10

sqrt(1/p^2) #sd

## [1] 10

2.1.3 c. Binomial

Binomial pmf is \(\begin{pmatrix}n\\ p\end{pmatrix}p^{k}(1-p)^{n-k}\). \(E[X]=np\) and \(V[X]=np(1-p)\).

The probability that the machine will fail after 8 years is \(0.4304672\).

The expected value is \(0.8\) and the standard deviation is \(0.8485281\).

pbinom(0,n,p) #0 machine failures in 8 years

## [1] 0.4304672

n*p #E[X]

## [1] 0.8

sqrt(n*p*q) #sd

## [1] 0.8485281

2.1.4 d. Poisson

Poisson pmf is \(\frac{\lambda^{k}e^{-\lambda}}{k!}\) where \(k\) is the number of occurrences. \(E[X]=\lambda\) and \(V[X]=\lambda\).

As the average number of machine failure in every 10 years is 1, the average number of occurrences in 8 years will be 8/10.