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 I am using solution and explanation provided by this link.

https://math.dartmouth.edu/archive/m20f10/public_html/HW5Solutions.pdf

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

Since Y is the minimum value of Xi over all of the Xi’s, then in order to find the distribution function m(j) = P(Y = j), we will need to count the number of ways that we can assign X1, X2, …, Xn to values between j and k with at least one Xi being assigned to j and divide by the total number of possible ways to assign X1, X2, …, Xn 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..)

\(P(X=x)=(1-p)^{ x-1 }p\)

p1<-1/10
p2<-pgeom(8,p1,lower.tail = FALSE)
p2
## [1] 0.3874205
ev<-1/(1/10)
ev
## [1] 10
sd<-sqrt((1-p1)/p1^2)
sd
## [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.

\(1 - P(X <= 8)\)

p1<-1/10
p2<-pexp(8,p1,lower.tail = FALSE)
p2
## [1] 0.449329
ev<-1/(p1)
ev
## [1] 10
sd<-sqrt(1/(p1^2))
sd
## [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)

\(P(X=0)=B(8, 0.1)\)

p1<-1/10
p2<-pbinom(0, 8, p1)
p2
## [1] 0.4304672
ev<-8*p1
ev
## [1] 0.8
sd<-sqrt((8*p1)*(1 -p1))
sd
## [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.

lambda1<-.8
p2<-ppois(0, lambda=.8)
p2
## [1] 0.449329
ev<-lambda1
ev
## [1] 0.8
sd<-sqrt(lambda1)
sd
## [1] 0.8944272