Problem 7.1

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 .

The function for Y = Min(X1, X2,… Xn) = Xi, where Xi is the minimum value for all independent random variables We can model the function as a binomial distribution, such that:

Xi = successful trial X1 to Xi-1, Xi+1 to Xn = failed trials n = # of trials k = population size (total outcomes/trial) p = 1/k (SUCCESS) q = n-1/k (FAILURE)

The binomial distribution for Y is: Y = n!/(n-1)!1! * p^(1) * q^(n-1) Y = n * p * q^(n-1)

Y = n * (1/k)* (n-1/k)^(n-1) Y = n * (1/k) * (n-1)(n-1)/k(n-1) Y = n * (n-1)(n-1)/kn

Problem 7.2

knitr::include_graphics('7-2.png')

(a)

# Creating vectors and calculating prob for failure after 8 years

prob <- 1/10
1-sum(dgeom(0:7, prob))
## [1] 0.4304672

(b)

exp <- 1/10
pexp(8, exp, lower.tail = FALSE)
## [1] 0.449329
# Mean

1/exp
## [1] 10
# Standard Deviation

1/exp
## [1] 10

(c)

binom <- 1/10
n <- 8

dbinom(0,n,binom)
## [1] 0.4304672
# mean

binom*n
## [1] 0.8
# SD

sqrt(binom*(1-binom)*n)
## [1] 0.8485281

(d)

p <- 1/10
n <- 8
lambda <- n*p
dpois(0, lambda)
## [1] 0.449329
# mean
lambda
## [1] 0.8
# SD

sqrt(lambda)
## [1] 0.8944272