Question 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.

Solution:

P(Y<= y) = 1- P(Y>y) = 1 - P(min{X1, X2, X3… Xn} > y) = 1 - P(X1 > y, X1 > y, … , Xm > y) = 1 - ((k-y)/k)^n

so with substitution:

P(Y<= y-1) = 1 - ((k-y + 1)/k)^n

Question 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.).

Part 1

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..)

# probability

pgeom(8, 0.1, lower.tail = F)
## [1] 0.3874205
# expected value

1/(.1)
## [1] 10
# Standard deviation

sqrt(1/(.1^2))
## [1] 10

Part 2

What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as an exponential.

# probability

pexp(8, 0.1, lower.tail = F)
## [1] 0.449329
# expected value

np = 8*.1
np
## [1] 0.8
# standard deviation

sd = sqrt(np*.9)
sd
## [1] 0.8485281

Part 3

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)

# probability

pbinom(0, 8, 0.1)
## [1] 0.4304672
# expected value

np=8*0.1

np
## [1] 0.8
# stndard deviation

sd = sqrt(8*0.1*0.9)

sd
## [1] 0.8485281

Part 4

What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as a Poisson.

# Probability

x <- 8
mu <- 10

dpois(x, mu)
## [1] 0.112599
# expected vlue

mu
## [1] 10
# standard deviation

sqrt(mu)
## [1] 3.162278