Question 1:

Let \(X_1, X_2, . . . , X_n\) 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 \(X_i\)’s. Find the distribution of \(Y\).

\(F(X_i; Y, k) = \frac{X_i - Y + 1}{k - Y + 1}\)

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.)

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 <- 0.1
expected <- (1 - p) / p
st_dev <- round(((1 - p) / p^2)^0.5, 3)
x = 8
p_after8yrs <- round(pgeom(q = x, prob = p, lower.tail = FALSE), 3)

\(P(X \ge 9) = 0.387\)

Expected: \(9\)

Standard Deviation: \(9.487\)

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.

lambda <- 0.1
expected <- 1 / lambda
st_dev <- (1 / lambda^2)^0.5
x = 8
p_after8yrs <- round(pexp(q = x, rate = lambda, lower.tail = FALSE), 3)

\(P(X > 8) = 0.449\)

Expected: \(10\)

Standard Deviation: \(10\)

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 <- 0.1
successes = 0
trials = 8
expected <- trials * p
q = 1 - p
st_dev <- round((trials * p * q)^0.5, 3)
p_after8yrs <- round(pbinom(q = successes, size = trials, prob = p), 3)

\(P(k = 0) = 0.43\)

Expected: \(0.8\)

Standard Deviation: \(0.849\)

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.

r = 0.1
t = 8
k = 0
lambda = r * t
expected = lambda
st_dev = round(lambda^0.5, 3)
p_after8yrs <- round(ppois(q = k, lambda = lambda), 3)

\(P(k = 0) = 0.449\)

Expected: \(0.8\)

Standard Deviation: \(0.894\)