homework7

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 .

Steps to find the distribution of \(Y\):

1. Define the Probability Mass Function (PMF) of \(Y\):

   The PMF of \(Y\), denoted as \(P(Y = y)\), where \(y\) ranges from 1 to \(k\), represents the probability that \(Y\) takes on each possible value.

2. Calculate \(P(Y = y)\) for \(y = 1\):

   For \(y = 1\), the probability that \(Y\) is equal to 1 is given by:

   \[ P(Y = 1) = 1 - \left(1 - \frac{1}{k}\right)^n \]

3. Calculate \(P(Y = y)\) for \(1 < y \leq k\):

   For \(1 < y \leq k\), the probability that \(Y\) is equal to \(y\) is given by:

   \[ P(Y = y) = \left(1 - \frac{y - 1}{k}\right)^n - \left(1 - \frac{y}{k}\right)^n \]

   This represents the probability that at least one of the \(X_i\)’s equals \(y\), multiplied by the probability that all other \(X_i\)’s are greater than \(y - 1\).

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..)

p <- 1/10
n <- 8

prob_of_fail <- pgeom(n - 1, prob = p, lower.tail = FALSE)
prob_of_fail

## [1] 0.4304672

Expected value

print(1/p)

## [1] 10

Standard deviation

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

## [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.

p <- 1/10
prob_fail_after_8_years <- 1 - pexp(8, rate = p)

print(prob_fail_after_8_years)

## [1] 0.449329

Expected value

print(1/p)

## [1] 10

Standard deviation

standard_deviation <- 1/p
standard_deviation

## [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)

dbinom(0, size = 8, prob=p)

## [1] 0.4304672

Expected value

expected_value <- n * p
expected_value

## [1] 0.8

Standard deviation

standard_deviation <- sqrt(n * p * (1 - p))
standard_deviation

## [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.

avg <- 8 * (1/10)
prob <- 1 - ppois(0,avg)
prob

## [1] 0.550671

Expected value(same as mean)

avg

## [1] 0.8

Standard deviation(sqrt of the avg)

standard_deviation <- sqrt(avg)
standard_deviation

## [1] 0.8944272

References

geometric

binomial

exponetial

poisson