Stat 231 Exam 1 Fall 2011 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed ATTENTION! Incorrect numerical answers unaccompanied by supporting reasoning will receive NO partial credit. Correct numerical answers to difficult questions unaccompanied by supporting reasoning may not receive full credit. SHOW YOUR WORK/EXPLAIN YOURSELF! Completely absurd answers (that fail basic sanity checks but that you don't identify as clearly incorrect) may receive negative credit. 1
1. Random variables X and Y have jointly continuous distribution with joint density 1 if 0 y 1 and y x y 1 f x, y 0 otherwise The region where the density is positive is indicated below. a) Find the marginal pdf of the random variable X, g x. (Be sure to sure to state carefully for what values of x any formulas you write hold.) b) Find P X Y 1. 6 pts c) Set up completely (but you need NOT evaluate) a double integral giving E XY. 2
2. In a production facility, there is occasional need for overtime labor on Saturdays. On a given Saturday, the number of workers needed is N and the number available is A. The marginal pmf's of these variables are indicated in the table below. Suppose the variables are independent. n 0 1 2 2.2 a 1.4 0.4.3.3.4 a) Fill in the 9 joint probabilities for the variables N and A in the table above. Use your table to answer the questions on the rest of this page. (If you don't know how to answer a), put any 9 positive probabilities summing to 1.0 into the table and go on.) b) Evaluate the probability that more workers are needed than are available. (This is P N A.) c) Evaluate the conditional probability that 2 workers are needed, given that more workers are needed P N 2 N A.) than are available. (This is The labor contract specifies that if more workers are needed than are available, the union pays a penalty equal to half of one day's cost for a worker. So, the company's overtime costs (in units of the weekday labor cost for one worker) for a given Saturday are Cost 1.5 number of workers working contract penalty (where the contract penalty is either 0 or.5 ). d) Evaluate E Cost, the mean overtime cost for a given Saturday. 3
3. Roughly speaking, if the number of arrivals at a small Emergency Room during one shift is 10 or more, there will be excessive waiting times required on some cases during that shift. Suppose that the number of arrivals during the 1 st shift on a weekday can be modeled as Poisson with mean 7. a) Evaluate the probability that on Monday, some cases require excessive waiting times during the 1 st shift (this means that there are at least 10 arrivals). Use your answer to a) in the following two questions. If you were unable to do a), you may use the incorrect value.3 in its place. b) Evaluate the probability that there are excessive waiting times on more than one day of the upcoming 5-day workweek. (Assume that the numbers of arrivals on the 5 days are independent.) 6 pts c) Evaluate the mean number of days in the upcoming workweek where excessive waiting times are experienced in this Emergency Room during the 1 st shift. 4
4. Suppose the actual diameters of a batch steel cylinders are normally distributed with mean 2.5025 inch and standard deviation.005 inch. a) Find the probability that a randomly selected cylinder from this batch has diameter between 2.495 inch and 2.505 inch. b) Suppose that diameters of cylinders are measured with a digital gauge that rounds the actual diameter to the nearest.01 inch (but is otherwise a perfect measuring device). So, for example, a steel cylinder with actual diameter between 2.495 inch and 2.505 inch will be measured as 2.50 inch (and the answer to a) is the fraction of cylinders from the batch that the digital gauge will measure as 2.50 ). If a cylinder is selected at random from the batch and Y is the diameter as measured by the digital gauge, a pmf for it is mostly specified below. Add your answer to a) above to the table and finish filling it in. Then find EY and compare it with the mean diameter of the cylinders 2.5025. y f y 2.52 2.51.3023 2.50 2.49.0666 2.48.0002 EY Compared to the actual mean diameter? 5
5. A random variable U has a uniform distribution on the interval 1,1 if it has pdf For such a random variable, EU k 1 if 1 u 1 f u 2 0 otherwise k 1 0 for odd k and E U for even k. (Take this as given.) k 1 a) Use the information above and the laws of expectation and variance to evaluate 2 VarU. b) What is the "propagation of error" approximation to 2 VarU. (Give a numerical value.) 6