91Ó°ÊÓ

Let \(n\) be a positive integer. Let \(S\) be the set of integers between 1 and \(n\). Consider the following process: We remove a number from \(S\) at random and write it down. We repeat this until \(S\) is empty. The result is a permutation of the integers from 1 to \(n\). Let \(X\) denote this permutation. Is \(X\) uniformly distributed?

Let \(X\) be a random variable which can take on countably many values. Show that \(X\) cannot be uniformly distributed.

Let \(U, V\) be random numbers chosen independently from the interval [0,1] with uniform distribution. Find the cumulative distribution and density of each of the variables (a) \(Y=U+V\) (b) \(Y=|U-V|\)

Let \(U, V\) be random numbers chosen independently from the interval [0,1] . Find the cumulative distribution and density for the random variables (a) \(Y=\max (U, V)\) (b) \(Y=\min (U, V)\)

(Ross \(^{11}\) ) An expert witness in a paternity suit testifies that the length (in days) of a pregnancy, from conception to delivery, is approximately normally distributed, with parameters \(\mu=270, \sigma=10 .\) The defendant in the suit is able to prove that he was out of the country during the period from 290 to 240 days before the birth of the child. What is the probability that the defendant was in the country when the child was conceived?

You are presented with four different dice. The first one has two sides marked 0 and four sides marked \(4 .\) The second one has a 3 on every side. The third one has a 2 on four sides and a 6 on two sides, and the fourth one has a 1 on three sides and a 5 on three sides. You allow your friend to pick any of the four dice he wishes. Then you pick one of the remaining three and you each roll your die. The person with the largest number showing wins a dollar. Show that you can choose your die so that you have probability \(2 / 3\) of winning no matter which die your friend picks. (See Tenney and Foster. \(\left.^{8}\right)\)