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.

http://mathworld.wolfram.com/UniformDistribution.html

The uniform distribution is defined as: \[f(x) = \begin{cases} 0, & \text{if } x < a \\ \frac{1}{b-a}, & \text{if } a \leq x \leq b \\ 0, & \text{if } x > b \\ \end{cases}\]

The Cumulative Distribution Function: \[F(x)=\int_a^x\!\frac{1}{b-a}dw=\bigg[\frac{w}{b-a}\bigg]_a^x=\frac{x-a}{b-a}\]

\[f(x) = \begin{cases} 0, & \text{if } x < a \\ \frac{x-a}{b-a}, & \text{if } a \leq x \leq b \\ 1, & \text{if } x > b \\ \end{cases}\]

In our case, x=x, a=1, b=k

\[f(x) = \begin{cases} 0, & \text{if } x < 1 \\ \frac{x-1}{k-1}, & \text{if } 1 \leq x \leq k \\ 1, & \text{if } x > k \\ \end{cases}\]

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 <- .10 # failure rate
q <- .90 # survival rate

# geometric distribution
pgeom(8, p, lower.tail = FALSE)
## [1] 0.3874205
EV <- 1 / p
EV
## [1] 10
sd <- sqrt(q / p^2)
sd
## [1] 9.486833
  1. What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as an exponential.
n <- 8

# exponential distribution
pexp(8, p, lower.tail = FALSE)
## [1] 0.449329
EV <- 1 / p
EV
## [1] 10
sd <- sqrt(1 / p^2)
sd
## [1] 10
  1. 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)
n <- 8

# binomial distribution
pbinom(0, 8, p, lower.tail = TRUE)
## [1] 0.4304672
EV <- n * p
EV
## [1] 0.8
sd <- sqrt(n * p * q)
sd
## [1] 0.8485281
  1. What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as a Poisson.
n <- 8

# Poisson distribution
ppois(0, p, lower.tail = TRUE)^n
## [1] 0.449329
EV <- 10
EV
## [1] 10
sd <- sqrt(EV)
sd
## [1] 3.162278