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 .

As we can see above this distribution is long tail right skewed.

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..)

## [1] "The probability that the machine will fai after 8 years is:-  0 4"
## [1] "Expected Value:-  10"
## [1] "Standard deviation is:-  9.4868"

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] "The probability that the machine will fail after 8 years is:-  0.4493"
## [1] "Expected Value:-  10"
## [1] "Standard deviation is:-  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)

## [1] "The probability that the machine will fail after 8 years is:-  0.4305"
## [1] "Expected Value:-  0.8"
## [1] "Standard deviation is:-  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.

## [1] "The probability that the machine will fail after 8 years is:-  0.4493"
## [1] "Expected Value:-  0.8"
## [1] "Standard deviation is:-  0.8944"