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 Xis. Find the distribution of Y

P(Y=min{X1,X2,…Xn})= number of ways for Y=min of Xinumber of possible outcomes/Number of possible outcomes:

for each Xi∈[1,k], there are kpossibilities, total number of possible outomes (denominator of P(Y=min{X1,X2,…Xn})) is kn

Suppose, Y= 1, we need to exclude outcomes where Xi≠1 so that each Xi∈[2,k] The number of ways for Y=min{X1,X2,…Xn}is kn−(k−1)n

Similarly when Y= 2, we not only need to exclude outcomes where xi≠2, but also need to exclude outcomes where Xi≤1 The number of ways for Y=min{X1,X2,…Xn}is kn−(k−2)n−[kn−(k−1)n]= (k−1)n−(k−2)n To generalize the answer, when Y=a, since all other terms are eliminated, the number of ways for Y=min{X1,X2,…Xn} is (k−a+1)n−(k−a)n

As a result, P(Y=min{X1,X2,…Xn})= number of ways for Y=min of Xi/number of possible outcomes= (k−a+1)n−(k−a)nkn

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 expected 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.)

Geometric

(geom<-round(pgeom(8, .1, lower.tail = FALSE),2)) #probability that the machine will fail after 8 years 
## [1] 0.39
(exvalue1<-1/.1) #expected value
## [1] 10
(sd1<-round(sqrt((1-.1)/(.1^2)),2)) #standard deviation
## [1] 9.49

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.

Exponential

(expon<-round(pexp(8, .1, lower.tail = FALSE),2)) #the probability that the machine will fail after 8 years
## [1] 0.45
(exvalue2<-1/.1) #expected value
## [1] 10
(sd2<-round(sqrt((1-.1)/(.1^2)),2)) #standard deviation
## [1] 9.49

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)

Binomial

(round(pbinom(0, 8, .1 ),2)) #the probability that the machine will fail after 8 years
## [1] 0.43
(exvalue3<-.1*8) #expected value
## [1] 0.8
(sd3<-round(sqrt(8*.1*(1 - .1)),2)) #standard deviation
## [1] 0.85

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.

Poisson

(round(pexp(8, .1, lower.tail = FALSE),2)) #the probability that the machine will fail after 8 years
## [1] 0.45
(exvalue4<-.1*8) #expected value
## [1] 0.8
(sd4<-round(sqrt(8*.1),2))  #standard deviation
## [1] 0.89