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.

Solution (maybe)

\(P(X = 1) =(k^n - (k - 1) ^n)/(k^n)\)

Continuing:

\(P(X = 3) =(k^n - (k - 3) ^n)/(k^n)\)

We generalize:

\(P(X = m) =(k^n - (k - m) ^n)/(k^n)\)

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 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.)

Solution

Using the geometric model: \(Pr(X = k) = (1-p)^{k-1}p\)

Let’s take the cumulative odds of the machine having failed in the first 8 years and then subtract it from 1. With the geometric distribution, the odds of a failure every year are lower due to the fact that they also include the odds that it did not fail the previous years.

1 - pgeom(8, (1/10))
## [1] 0.3874205

Expected value: 10 years - flip the fraction

1/(1/10)
## [1] 10

Standard Deviation:

For a geometric distribution, this is \(\sqrt{(1-p)} / p\)

sqrt(1-.1)/.1
## [1] 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.

Solution

Using the exponential model:

Exponential model: \(P(X \leqslant x) = 1 -e^{-mx}\)

We’ll get the complement.

1 - pexp(8, (1/10))
## [1] 0.449329

Expected value: Still 10

1/(1/10)
## [1] 10

Standard deviation: \(\sqrt{(1/p^2)}\)

sqrt(1/(1/10)^2)
## [1] 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)

Solution

Modelling as a binomial: \(p^x(1-p)^{n-x}\)

We’ll look at getting no success outcomes in 8 years.

pbinom(0,8,.1)
## [1] 0.4304672

Expected value (first success): In this context, that means .8 expected failures in 8 years.

.1 * 8
## [1] 0.8

Standard deviation:

sqrt(8*.1*(1-.1))
## [1] 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.

Solution

Let’s get Poisson: \((e^-{\lambda} \lambda^{x}) / x!\)

ppois(0, 8/10)
## [1] 0.449329

Expected value:

8/10
## [1] 0.8

Standard deviation:

sqrt(8/10)
## [1] 0.8944272