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 .

For 1 ≤ j ≤ k, m(j) = ((k−j+1)n−(k−j)n)/k^n

Since Y is the minimum of the Xs, 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..)

Using a geometric distribution:

f(x)=P(X=x)=(1−p)^x−1 * p P(X=8)=(9/10)^7 ∗ 1/10

dgeom(8,0.1)
## [1] 0.04304672

expected Value= 1/p

1/(1/10)
## [1] 10

standard deviation= √(1−p)/p^2

sqrt((1 - (1/10))/((1/10)^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.

Exponential Distribution

rate= 1/10

Failure after 8 years: 1-P(X<=P)

1-pexp(8, 0.1)
## [1] 0.449329

Expected value = 1/0.1

1/0.1
## [1] 10

Standard Deviation = 1/(0.1)^2

1/(0.1)^2
## [1] 100

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)

Binomial Distribution

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

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

Expected Value = n * p

8*0.1
## [1] 0.8

Standard deviation = √(np∗(1−p)

sqrt((8 * 0.1) * (1 - 0.1))
## [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.

Poisson Distribution

X ~ P(0.8)

P(X=0)

ppois(0,0.8)
## [1] 0.449329

Expected value = 0.8

Standard Deviation = √(expected value)

sqrt(0.8)
## [1] 0.8944272