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 .

Say j is the minimum integer value, then we need to find how many ways we can distribute all the variables over k-j integers. As we distribute the n variables, they have k^n options to be distributed to. This is the denominator. Since j is the minimum, there are truly k-j spaces that we are concerned wtih. But what if j=1, then there are k^n - (k-1)^n spaces. If j=2, then there are (k-1)n-(k-2)n spaces. If j=j, there are (k-j+1)n-(k-j)n spaces. So the distribution is ((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..) P(X>8)=((.9)^8)(.1)=.04304672 EV=1/.1 = 10 years stdev = 9.486833 years

  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(X>8)=((e)^(-.8))=.04304672 EV=1/.1 = 10 years stdev =1/.1 = 10 years

  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) P(X>8)=(8C0)(.9^8)*(.1^0)=.4304672 EV=.1 years stdev=(.1)^.5 = .3162278 years

  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. P(X>8)=((.18)(e(-.1)))/(8!)=2.2441404e-13 EV= .1 years stdev=(.1)^.5 = .3162278 years