Stat 425 Problems due 4/11/06 Chapter 7: 9a (there can be more than one ball in an urn), 11, 32, 34 A) An environmental engineer measures the amount (by weight) of particulate pollution in air samples of a certain volume collected over the smokestack of a coal-operated power plant. Let Y1 denote the amount of pollutant per sample collected when the cleaning device on the stack is not operating and let Y2 denote the amount of pollutant per sample collected under the same environmental conditions when the cleaning device is operating. It is observed that the <a href="/keyword/joint-probability-density-function/" ><a href="/keyword/joint-probability-density/" >joint probability density</a> function</a> of Y1 and Y2 can be modeled by fY1 Y2 (y1 , y2 ) = K 0 if 0 y1 2; 0 y2 1; 2y2 y1 ; elsewhere. (That is, Y1 and Y2 are uniformly distributed over the region inside the triangle bounded by y1 = 2, y2 = 0 and 2y2 = y1 .) 1. Find the value of K that makes this function a <a href="/keyword/joint-probability-density-function/" ><a href="/keyword/joint-probability-density/" >joint probability density</a> function</a> . 2. Find P (Y1 3Y2 ). (That is, find the probability that the cleaning device will reduce the amount of pollutant by one-third or more.) 3. If the cleaning device is operating, find the probability that the amount of pollutant in a given sample will exceed 0.5. 4. Given that the amount of pollutant in a sample taken with the cleaning device operating is 0.5, find the probability an non-operating cleaning device would have lead to a pollutant level exceeding 1.5. B) Let Y1 and Y2 denote the proportions of two different types of components in a sample from a mixture of chemicals used as an insecticide. Suppose Y1 and Y2 have the joint density function given by fY1 Y2 (y1 , y2 ) = 2, 0 y1 1; 0 y2 1; 0 y1 + y2 1 0, elsewhere. (Note that Y1 + Y2 1 because the random variables denote proportions within the same sample.) 1. Find P [Y1 3/4, Y2 3/4]. 2. Find P [Y1 1/2|Y2 1/4] (use definition for conditional probabilities of events here). 3. (d) Find P [Y1 1/2|Y2 = 1/4]. 1 The problems below are for extra practice when studying for the exam. Chapter 7: 8(hint: any two people are friends with probability = p), 12a, 36,44 A) Three balanced coins are tossed independently. One of the variables of interest is Y1 = the number of heads. Let Y2 denote the amount of money won on a side bet in the following manner. If the first head occurs on the first toss, you win $1. If the first head occurs on toss 2 or on toss 3 you win $2 or $3, respectively. If no heads appear, you lose $1 (that is, win -$1). 1. Find the joint probability mass function for Y1 and Y2 . 2. What is the probability that less than three heads occur and you win $1 or less? (That is, find FY1 Y2 (2, 1).) 3. Derive the probability mass function for your winnings on the side bet. 4. What is the probability that you obtained three heads given that you won $1 on the side bet? B) Let Y1 and Y2 denote the proportions of time, out of one workday, that employees I and II, respectively, actually spend performing their assigned tasks. The <a href="/keyword/joint-probability-density-function/" ><a href="/keyword/joint-probability-density/" >joint probability density</a> function</a> of Y1 and Y2 is fY1 Y2 (y1 , y2 ) = 1. Find P (Y1 &lt; 1/2, Y2 &gt; 1/4). 2. Find P (Y1 + Y2 1). 3. Find the probability density function of Y1 . Find the probability density function of Y2 . 4. Find P (Y1 1/2|Y2 1/2). 5. If employee II spends exactly 50% of the day on assigned duties, find the probability that employee I spends more than 75% of the day on assigned duties. C) A supermarket has two customers waiting to pay for their purchases at counter I and one customer waiting to pay at counter II. Let Y1 and Y2 denote the numbers of customers who purchase more than $50 of groceries at the respective counters. Suppose Y1 and Y2 are independent binomial random variables, with the probability of a customer spending more than $50 equal to .2 for counter I and .3 for counter II. 1. Find the joint probability mass function for Y1 and Y2 . 2 y1 + y2 , 0 y1 1; 0 y2 1 0, elsewhere. 2. Find the probability that no more than one of the three customers spends in excess of $50. D) Two telephone calls come into a switchboard at random (uniformly) times in a fixed 1-hour period. Assume the calls are made independently of one another. 1. What is the probability that both calls are made in the first half hour? 2. What is the probability that the calls are made within 5 minutes of each other? 3