CUNY MSDS DATA 605 HW 7

library(visualize)

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

\(K^n\) represents the total num of combinations. \((k-1)^n\) represents the combinations where none of the Xi are equal to 1.

\[P(X = 1) = \frac{k^n - (k-1)^n}{k^n}\]

\[P(X = 2) = \frac{(k-2+1)^n - (k-2)^n}{k^n}\]

\[P(X = y) = \frac{(k-y+1)^n - (k-y)^n}{k^n}\]

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..)

This can be achieved in R using the pgeom function, or by using the visualize package.

pgeom(8, 0.1, lower.tail = F)

## [1] 0.3874205

visualize.geom(8, prob = 0.1, section = "upper")

\[E(X) = 1/\lambda = 1/0.1 = 10\] \[\sigma^2 = \sqrt{1/\lambda^2} = 1/\lambda = 10\]

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.

This can be achieved in R using the pexp function, or by using the visualize package.

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

## [1] 0.449329

visualize.exp(stat = 8, theta = 0.1, section = "upper")

\[E(X)=np= 8 \times 0.1 = 0.8\] \[\sigma^2 = \sqrt{npq} = \sqrt{8 \times 0.1 \times 0.9} \approx 0.8485\]

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

visualize.binom(stat = 0, size = 8, prob = 0.1, section = "lower")

\[E(X)=np= 8 \times 0.1 = 0.8\]

\[\sigma^2 = \sqrt{npq} = \sqrt{8 \times 0.1 \times 0.9} \approx 0.8485\]

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.

\[P(X = x) = \frac{e^{-\mu}\mu^x}{x!}\]

This can be achieved in R using the dpois function, or by using the visualize package.

x <- 8
mu <- 10

dpois(x, mu)

## [1] 0.112599

visualize.pois(stat = c(x,x), lambda = mu, section = "bounded")

## Supplied strict length < 2, setting inequalities to  equal to  inequality.

Expected Value = \(\mu\) Standard deviation = \(\sqrt{\mu}\)

mu

## [1] 10

sqrt(mu)

## [1] 3.162278