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 .

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

  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..)
q <- 0.9
p <- 0.1
q^8 * sum((q^(0:100)) * p)
## [1] 0.4304569

\[ P(T > k) =\sum_{j=k+1}^{\inf} q^{j-1}p = q^k(p + qp + q^2p + · · ·)=q^k\\ P(T > 8) =\sum_{j=8+1}^{\inf} (.9)^{9-1}(.1) = (.9)^8(.1 + .9*.1 + .9^2*.1 + · · ·)=.9^8\\ P(T > 8) \approx 0.4305\\ \]

\[ E(T) =\frac{p}{(1 − q)^2}=\frac{p}{p^2}=\frac{1}{p}\\ E(T) =\frac{.1}{(1 − .9)^2}=\frac{.1}{.1^2}=\frac{1}{.1}\\ E(T) = 10\\ \]

The expected value is 10 years.

\[ \sigma=\sqrt{\frac{q}{p^2}}\\ \sigma=\sqrt{\frac{.9}{.1^2}}\\ \sigma \approx 9.49\\ \]

The standard deviation is approximately 9.49.

  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.
1-pexp(8,.1)
## [1] 0.449329

\[ 1-F(8) = 1-P(T ≤ 8)\\ = 1-\int_{0}^{8}0.1e^{−0.1t} \, dt\\ =1- (1 − e^{(-0.1)(8)})\\ =e^{\frac{-4}{5}}\\ \approx 0.4493\\ \]

\[ E(X)=\frac{1}{\lambda}\\ E(X)=\frac{1}{.1}\\ E(X)=10\\ \] The expected value is 10 years.

\[ \sigma=\frac{1}{\lambda}\\ \sigma=\frac{1}{.1}\\ \sigma=10\\ \] The standard deviation is also 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)
pbinom(0,8,.1)
## [1] 0.4304672

\[ b(n, p, k) =\binom{n}{k} p^kq^{n-k}\\ b(8, .1, 0) =\binom{8}{0}.1^0.9^{8-0}\\ b(8, .1, 0) =1 \cdot 1 \cdot .9^{8}\\ b(8, .1, 0) \approx 0.4305\\ \]

\[ E(X)=np\\ E(X)=8*.1\\ E(X)=0.8\\ \] There are 0.8 successes expected in 8 years.

\[ \sigma = \sqrt{np(1-p)}\\ \sigma = \sqrt{8*0.1*0.9}\\ \sigma \approx 0.85\\ \] The standard deviation is approximately 0.85.

  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.
ppois(0, .8)
## [1] 0.449329

\[ P(Y = k) = e^{−λt}\frac{ (λt)^k}{k!}\\ P(Y = 0) = e^{−.1(8)}\frac{ (.1(8))^0}{0!}\\ P(Y = 0) = e^{−.8}\\ P(Y = 0) \approx 0.4493\\ \]

\[ E=\lambda=10\\ \]

The expected value is 10 years.

\[ \sigma = \sqrt{\lambda}\\ \sigma = \sqrt{10}\\ \sigma \approx 3.16\\ \]

The standard deviation is approximately 3.16.