HW #7 - Distribution & Densities,Expected Value & Variance

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.

Answer:

Y = Minimum(Xis)

Since each random variable Xi has k possibilities, the total possible number of assignment of the whole number collection is k^n

Let’s analyze the situation when Y=1. To get the number of possibilties that Y=1, we can exclude the number of possibilities that Xi!=1 out of k^n.

Thus, the number of possibilities is k^n - (k-1)^n.
P(Y=1) = (k^n - (k-1)^n) / k^n

Similarly, when Y=2, we need to exclude the number of possibilities that Xi=2 and Xi<=1 from k^n.

P(Y=2) =[ k^n - (k-2)^2 - [(k^n - (k-1)^n)] ]/ k^n

= ((k-1)^n - (k-2)^n)/k^n

When Y = a, then P(Y=a) = ((k-a+1)^n - (k-a)^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 fai 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..)
library('stats')
pfail = 1/10 
psucc = 1-pfail 
n = 8 

result = 1- pgeom(n-1,pfail)

The probability that the machine will fai after 8 years is 0.4304672

expect_value = 1/pfail

Expected Value: E(X) = 10

sd = sqrt(psucc/pfail^2)

Standard deviation is 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.
n= 8
lambda = 1/10 

p_expo = pexp(n,lambda,lower.tail=FALSE)

The probability that the machine will fail after 8 years is 0.449329

expect_value = 1/lambda

Expect vaue is 10

sd = sqrt(1/lambda^2)

Standard Deviation is 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)
b= 8 
p = 1/10 
q = 1- p 
k = 0

p_binomial = dbinom(k,n,p)

The probability that the machine will fail after 8 years is 0.4304672

expect_value = n*p

Expected value is 0.8

sd = sqrt(n*p*q)

Standard deviation is 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
lambda <- 8/10
k <- 0

p_poison= ppois(0,lambda = .8 )

The probability that the machine will fail after 8 years is 0.449329

expect_value = 8/10

Expected value is 0.8

sd = sqrt(expect_value)

Standard deviation is 0.8944272