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.

\[For 1≤j≤k, m(j)= \frac{(k-j+1)^n-(k-j)^n}{k^n}\]

  1. 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..)

*** Probability:*** \[P(X=x) = (1-p)^{n-1}*p\]

p_8yrs = ((.9)^(7))*0.1
p_8yrs
## [1] 0.04782969

*** Expected Value:*** \[E[X] = \frac{1}{p}\]

EV = 1/0.1
EV
## [1] 10

*** Standard Deviation:*** \[sd = \sqrt\frac{1-p}{p^2}\]

sd = sqrt((1-0.1)/(0.1^2))
sd
## [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.

*** Probability:*** \[P(X>=8yrs) = e^\frac{-k}{u}\]

p_8yrs = exp(-8/10)
p_8yrs
## [1] 0.449329

*** Expected Value:***

\[E[X] = u = \frac{1}{\lambda} = 10, where, \lambda = \frac{1}{10}\]

EV = 1/0.1
EV
## [1] 10

*** Standard Deviation:*** \[sd = \sqrt\frac{1}{\lambda^2}\]

sd = sqrt(1/(0.1^2))
sd
## [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)

*** Probability *** \[P(X>8) = 1-p^x(1-p)^{n-x}\]

prob = 0.1^0*0.9^8
prob
## [1] 0.4304672

*** Expected Value: *** \[E[X] = np\]

EV = 8*0.1
EV
## [1] 0.8

*** Standard Deviation *** \[sd = \sqrt(n*p*q)\]

sd = sqrt(8*0.1*0.9)
sd
## [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.

*** Probability ***

\[P(X=8) = \frac{\lambda^xe^{-\lambda}}{x!}\] \[Where\] \[\lambda = \frac{np}{t} = \frac{8*0.1}{1} = 0.8\] \[x = 8\]

p_pois = (0.8^8)*exp(-0.8/8)
p_pois
## [1] 0.1518065

*** Expected Value:***

\[E[X] = \lambda = 0.8\]

*** Standard Deviation:*** \[sd = \sqrt\lambda\]

sd = sqrt(0.8)
sd
## [1] 0.8944272