# library(gtools)
library(matlib)
  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 .

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

  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..)

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

p=1/10
x=8
n=9
(prob_8y = (1-p)^(n-1) * p)
## [1] 0.04304672
(dgeom(x,p))
## [1] 0.04304672

\[ Expected\ Value\ Formula:\ E(X)=\frac{1}{p}\ or\ p^{-1} \]

p=1/10
(EV=1/p)
## [1] 10
(EV=p^-1)
## [1] 10

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

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

\[ Probability\ as\ Exponential:P(X>=8)=e^{\frac{-k}{\mu } } \]

k=8
lambda=1/10
mu=1/lambda
(prob_8y=exp((-k/(mu))))
## [1] 0.449329

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

(EV=mu)
## [1] 10

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

(sd=sqrt(1/lambda^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)

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

(n=10)
## [1] 10
(x=8)
## [1] 8
(choose(n,x)*p^x*((1-p)^(n-x)))
## [1] 3.645e-07

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

(n*p)
## [1] 1

\[ Standard\ Deviation\ Formula: \sigma =\sqrt{npq} \ where\ q=1-p \]

q=1-p
(sd = sqrt(n*p*q))
## [1] 0.9486833
  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. \[ P(X=8)=\frac{{}\lambda^{x} e^{-\lambda }}{x!} \ Where\ \lambda =\frac{np}{t} \]
t=1
x=8
(lambda=(n*p)/t)
## [1] 1
(p_pois=((lambda^x)*(exp(-lambda)))/factorial(x))
## [1] 9.123994e-06

\[ E[X]=\lambda \]

(lambda)
## [1] 1

\[ sd=\sqrt{\lambda } \]

(sd=sqrt(lambda))
## [1] 1