Data 602 - HW7

Question 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

If \({ K }^{ n }\) represents the total sum of the variables then \({ (k-1) }^{ n }\) can represent the varibles wheren none of Xi is equal to 1.

\[P(X=1)=\frac { { k }^{ n }-{ (k-1) }^{ n } }{ { k }^{ n } }\] \[P(X=2)=\frac { { (k-2+1) }^{ n }-{ (k-2) }^{ n } }{ { k }^{ n } }\] \[P(X=y)=\frac { { (k-y+1) }^{ n }-{ (k-y) }^{ n } }{ { k }^{ n } }\]

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.).

If the life expenticy for a failure is every 10 years, then we have an increasing presentage of 10% per year until we get to the 10th year.

p <- .10
initialsd <- round(sqrt((1-p)/p^2),4)
initialsd

## [1] 9.4868

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..)

q2a <- pgeom(8,p,lower.tail = FALSE)
q2a

## [1] 0.3874205

asd <- round(sqrt((1-p)/p^2),4)
asd

## [1] 9.4868

2. What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as an exponential.

q2b <- pexp(8, p, lower.tail = FALSE)
q2b

## [1] 0.449329

bsd<- sqrt(p^-2)
bsd

## [1] 10

3. 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)

pbinom(0,8,p)

## [1] 0.4304672

e <- exp(1)
csd <- round(e^(-p*8),4)
csd  

## [1] 0.4493

4. What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as a Poisson.

dpois(8,10)

## [1] 0.112599

round(sqrt(.8),4)

## [1] 0.8944