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 .

We have X1, X2, … to Xn, then total number of possibilities = kn The possibilities none of Xi equals to 1 is (k-1)^n

When X = 1 \[P(X=1) = (k^n-(k-1)^n)/k^n\] When X = 2 \[P(X=2) = ((k-2+1)^n)-(k-2)^n)/k^n\] When X = 3 \[P(X=2) = ((k-3+1)^n)-(k-3)^n)/k^n\] Therefore, we can get the Y distribution as \[P(X=y) = ((k-y+1)^n)-(k-n)^n)/k^n\]

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.).

  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 machine fails after 8 years.

prob_f<- 1/10
y<-8
prob_nf<- 1-prob_f
prob_f8y_geom<- 1-pgeom(y-1, prob_f)
prob_f8y_geom
## [1] 0.4304672

Expected value

ev<-1/prob_f
ev
## [1] 10

Standard deviation

sd<-sqrt(prob_nf/prob_f^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.

Probability machine fails after 8 years.

y<- 8
prob_f<- 1/10
prob_f8y_exp<- pexp(y, prob_f, lower.tail = F)
prob_f8y_exp
## [1] 0.449329

Expected value

ev<-1/prob_f
ev
## [1] 10

Standard deviation

sd<- sqrt(1/prob_f^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)
y<- 8
prob_f<- 1/10
prob_nf<- 1-prob_f
k<- 0
prob_f8y_bino<- dbinom(k, y, prob_f)
prob_f8y_bino
## [1] 0.4304672

Expected value

ev<- y * prob_f
ev
## [1] 0.8

Standard deviation

sd<-sqrt(y * prob_f * prob_nf)
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.
prob_f<- 1/10
y<- 8
k<- 0
lambda<- prob_f * y
lambda
## [1] 0.8
ppois(0, lambda )
## [1] 0.449329

Expected value

ev<-y * prob_f
ev
## [1] 0.8

Standard deviation

sd<- sqrt(ev)
sd
## [1] 0.8944272