Verify the following identity regarding the gamma function: ?( 1) = ( ). ? ? ?? ? 2) The length of time until the breakdown of an essential piece of equipment is important in the decision of the use...

Verify the following identity regarding the gamma function: ?( 1) = ( ). ? ? ?? ? 2) The length of time until the breakdown of an essential piece of equipment is important in the decision of the use of auxiliary equipment. Assume time to breakdown of a randomly chosen generator, Y, follows an exponential distribution with a mean of 15 days. (a) What is the probability a generator will break down in the next 21 days? (b) A company owns 7 such generators. Let X denote the random variable describing how many generators break down in the next 21 days. Assuming the breakdown of any one generator is independent of breakdowns of the other generators, what is the probability that at least 6 of the 7 generators will operate for the next 21 days without a breakdown? 3) Consider the following game: A player throws a fair die repeatedly until s/he rolls a 2, 3, 4, 5 or 6. In other words, the player continues to throw the die as long as s/he rolls 1s. When s/he rolls a “non-1,” s/he stops. (a) What is the probability the player tosses the die exactly three times? (b) What is the expected number of rolls needed to obtain the first non-1? (c) If the player rolls a non-1 on the first throw, the player is paid $1. Otherwise, the payoff is doubled for each 1 the player rolls before rolling a non-1. Thus, the player is paid $2 if s/he rolls a 1 followed by a non-1; $4 if s/he rolls two 1s followed by a non-1; $8 if s/he rolls three 1s followed by a non-1; etc. In general, if we let Y be the number of throws needed to obtain the first non-1, then the player rolls ?Y ?1? 1s before rolling his/her first non-1, and s/he is paid 1 2Y ? dollars. What is the expected amount paid to the player? (d) Suppose the game was modified and the player continues to throw the die until s/he rolls three non-1s. What is the probability the player tosses the die less than 5 times? 4) One of four different prizes was randomly put into each box of a cereal. If a family decided to buy this cereal until it obtained at least one of each of the four different prizes, what is the expected number of boxes of cereal that must be purchased? (Hint: Let X = # of boxes of cereal needed to get four prizes. Think of X as the sum of four random variables. Yi = # of boxes needed to get a prize you don’t already have, i = 1, 2, 3, 4.) 5) In a small pond there are 50 fish, 10 of which have been tagged. If a fisherman’s catch consists of 7 fish, selected at random and without replacement, what is the probability exactly two tagged fish are caught?