DATA_605_Assignment_7_Thonn

** DATA_605_Assignment_7_Thonn - Important Distributions **

# install libraries if needed
#install.packages("permutations")
#library(permutations)

#install.packages('gtools')
#library(gtools)

** Problem-1 **

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.

Answer-1:

P(Y=1)+P(Y=2)+???+P(Y=k)=1

a). assign 1 to k to X1, X2, . . . , Xn = kn

P(Y=1)=kn - (k - 1)nkn

b). assign 2 to k to X1, X2, . . . , Xn = kn

P(Y=2)=kn -(k - 2)n - [kn - (k-1)n]kn
=(k-1)n - (k-2)nkn

c). assign 3 to k to X1, X2, . . . , Xn = kn

P(Y=3)=kn-(k-3)n - [(k-1)n - (k-2)n] - [kn - (k-1)n]kn =(k-2)n - (k-3)nkn

d). assign j to k to X1, X2, . . . , Xn is (k - j)n

then-Answer-1:

P(Y=j)=(k-j+1)n - (k-j)nkn

these are the ways to assign X1,X2……Xn so that the minimum value is j.

** Problem-2 **

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..)

# Problem - 2a: 
#probability that machine will fail before 8 years
p_bef_8 <- pgeom(7,1/10)

#probability that machine will fail after 8 years
p_aft_8 <- round(1-p_bef_8,3)
p_aft_8

## [1] 0.43

# [1] 0.43

# expect 1 failure every 10 years
p <- 0.1
EV <- (1 - p)/p
EV

## [1] 9

# [1] 9

 sd <- sqrt((1 - p)/p^2)
 sd

## [1] 9.486833

# [1] 9.486833

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.

# Problem - 2b: 

x <- 8
lam <- .1

#probability that machine will fail after 8 years
p_exp_aft_8 <- round(pexp(8,.1,lower.tail = F),4)
p_exp_aft_8

## [1] 0.4493

# expect 1 failure every 10 years
x <- 8
lam <- .1
EV <- 1/lam
EV

## [1] 10

# [1] 10

sd <- sqrt(1/lam^2)
sd

## [1] 10

# [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)

pr <- 0.1
n <- 8
x <- 0

dbinom(0, 8, .1)

## [1] 0.4304672

# [1] 0.4304672


EV <- n * pr
EV

## [1] 0.8

# [1] 0.8

sd <- sqrt(n * pr * (1 - pr))
sd

## [1] 0.8485281

# [1] 0.8485281

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.

pr <- 0.1
t <- 8
x <- 0

lam <- t * pr
lam

## [1] 0.8

# [1] 0.8

dpois(x,lam)

## [1] 0.449329

# [1] 0.449329


EV <- lam
EV

## [1] 0.8

# [1] 0.8

sd <- round(sqrt(lam),4)
sd

## [1] 0.8944

# [1] 0.8944

** END **