Assignment7

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:

Since, the uniformly distributed integers range from 1 to k that means it is definitely greater than 1. And sum of any variables regardless of distribution would be a normal density function.

So, the distribution of Y will be a right skewed normal density funtion which starts from 0.

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..) Answer: Here, let x be the years until the machine fails.

According to the question, P(X = x) = (1 - p)^(x-1) * p = (9/10)^7 * 1/10

# we can use dgeom for this
y <- dgeom(8, 0.1)
y

## [1] 0.04304672

p <- 1/10
Expected_value <- 1/p
Expected_value

## [1] 10

st_dev = sqrt((1 - (1/10))/((1/10)^2))
st_dev

## [1] 9.486833

The probability that the machine will fail after 8 years is 9.48

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. Answer: Here, According to the question, Rate = 1/10

   Failure after 8 years: 1 - P(X <= 8)

y <- 1 - pexp(8,0.1)
y

## [1] 0.449329

p <- 1/10
Expected_value <- 1/p
Expected_value

## [1] 10

std_dev <- 1/((0.1)^2)
std_dev

## [1] 100

The probability that the machine will fail after 8 years is 100.

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) Answer: Here, According to the question,

y = dbinom(0, size=8, prob=0.1) 
y

## [1] 0.4304672

Expected_value <- 8 * 0.1
Expected_value

## [1] 0.8

std_dev = sqrt((8 * 0.1) * (1 - 0.1))
std_dev

## [1] 0.8485281

The probability that the machine will fail after 8 years is 0.84.

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. Answer: Here, According to the question,

y <- ppois(0,0.8)
y

## [1] 0.449329

# expected value = variance
Expected_value <- sqrt(0.8)
Expected_value

## [1] 0.8944272

The probability that the machine will fail after 8 years is 0.89