STAT 225 Fall 2015 --- Homework 4-------DUE at the beginning of class on Friday, Oct. 30, 2015. FOR ONLINE STUDENTSmust turn in via Blackboard by 11:59 pm on Oct. 30, 2015. Or turn in to the course coordinator's office by 5:00 pm. SHOW WORK!!!!! Full credit will not be given for answers only. NOTE: for any question asking you to determine a probability----you MUST write out a probability statement using proper notation!!!! Probabilities should be DECIMAL form and rounded to 4 decimal places. 1. The distance it takes George's car to stop at a certain speed once the brakes are applied varies evenly between 20 and 55 feet. a) Let S be the distance it takes for George's car to stop at this speed. Find the distribution, parameter(s), and support of S. b) What is the probability that it takes between 15 and 28 feet for the car to stop at this speed? c) Find the expected distance it will take George to stop and the standard deviation of the stopping distance. d) What is the 45th percentile of George's stopping distances at this speed? e) Suppose a squirrel decides to dart into the road as George is driving this speed, and when George finally sees the squirrel and applies the brakes, the squirrel is 31 feet away. What is the probability that the squirrel survives its encounter with George (i.e. that George stops before he hits the squirrel)? f) George is out driving at this speed when it begins to rain. George knows that when it rains, it will take a minimum of 27 feet to stop. If there is a stop sign that is 35 feet away, what is the probability that George stops in time? 2. Once the call button has been pushed, it takes 3.2 minutes on average for an elevator to arrive in the MATH building. Let E be the time an individual waits until the elevator arrives. a) What are the distribution, parameter(s), and support of E? b) What is the probability that an individual waits between 145 seconds and 190 seconds for an elevator to arrive? c) What is the mean waiting time? d) What is the median waiting time? e) Suppose a professor is running late and he only has four and a half minutes to get to class. Assuming that it takes him 2 minutes to ride down in the elevator and walk to his classroom, what is the probability that he makes it to class on time? f) The same professor is now waiting on an elevator to get back to his office. If he has already been waiting 1 min, what is the probability that he'll wait for more than a total of 210 seconds? g) Suppose it takes an individual 4 minutes to reach the first floor of the math building from the 8th floor when they take the stairs. If an individual takes the elevator, it will only take them 30 seconds to reach the first floor from the 8th floor. Mary has been waiting 2 minutes for an elevator already. Should Mary wait or take the stairs if she wants to reach the first floor more quickly? 3. A space probe encounters a planet capable of sustaining life on average every 2.6 lightyears. (Recall that a lightyear is a measure of distance, not time.) a) Let Z be the number of life-sustaining planets that the probe encounters in 10 lightyears. What are the distribution, parameter(s), and support of Z? b) What is the probability that the probe encounters at least 3 life-sustaining planets in 10 lightyears? c) The probe has just encountered a life-sustaining planet. What is the probability that it takes more than 4 lightyears to encounter the next life-sustaining planet? What distribution and parameter(s) are you using? d) Suppose the probe has not encountered a life-sustaining planet for 3.5 lightyears. Knowing this, what is the probability that it will take at most 10 lightyears to detect the next life-sustaining planet? e) The probe has encountered 11 life-sustaining planets in the last 15 lightyears. What is the probability that there are 3 life-sustaining planets in the first 2.5 lightyears of this 15-lightyear span? SKIP PROBLEM 4 4. Let a) b) c) d) e) f) 3( 2 0.25) 0 < 1 () = { ( 1)2 2 < 4 0 Find the value of c that makes the above pdf valid. Find the cumulative distribution function (i.e. cdf) of X. What is the probability that X is greater than 2.5? Given that X is greater than 1, what is the probability that X is less than 3? Find E[X]. Find the standard deviation of X. 5. Fill in the table below with respect to the random variable described in each scenario. Random Variable a) The arrival times of customers at a sandwich shop occur at average rate of 1 every 5 minutes. Let C represent the number of customers arriving between 1:30 pm and 1:45 pm. (Continued from above ... ) b) Knowing 6 customers arrived between 3:00 pm and 3:50 pm, let D represent the number of customers who arrived between 3:00 pm and 3:30 pm. c) Let T represent the time between customer service calls which occur at a rate of 3 per 2 minutes. d) Let L represent the length of time for oven-baked fish to cook, which varies evenly from 12-15 minutes. Distribution Parameter(s) Support 6. When pearl oysters are opened, pearls of various sizes are found. Suppose that each oyster contains a pearl with a diameter that varies evenly from 0 to 10 mm. Let D be the diameter of the pearl. a) Calculate the mean and standard deviation of the pearl diameter. b) What percent of pearls have diameter no more than 7.5 mm? c) What is the probability that a pearl has a diameter of exactly 5 mm? d) The pearl oyster farmer wants to know the value of the upper 20% of the pearl diameters. Find this value. e) Pearls with a diameter of at least 4 mm have commercial value. Calculate the probability that an oyster contains a pearl of commercial value. f) Suppose that a farmer retrieves ten oysters out of the water. Let the random variable P represent the number of them containing pearls of commercial value. Find the probability at least 8 of the oysters the farmer retrieves contain pearls of commercial value. Indicate what distribution(s) and parameter(s) you are using. Assume the oysters grow pearls independently of one another. 7. Let X be the current in a certain electrical circuit. The current is measured in amperes and has the following density function (i.e. pdf) () = { 0.075 + 0.2, 0 3 < < 5 a) b) c) d) Find the cdf of X. What is the probability that the current is equal to 3.3 amperes? Find the probability that X is more than 4 amperes. Given that the current is at least 4 amperes, what is the probability that the current exceeds 4.7 amperes? e) What is the mean current of this circuit? 8. A customer service center receives phone calls at a rate of 10 per hour. Let C be the number of phone calls received in a 4-hour shift. Let T be the amount time (in hours) of between 1st and 2nd calls. Assume that calls arrive independently. a) Name the distribution and parameter(s) of C. b) Name the distribution and parameter(s) of T. c) Find the probability that there are between 50 and 52 calls (inclusive) between 2 pm and 6 pm. d) A call came in to the customer service center at 2:05 pm, what is the probability that the next call comes in before 2:10 pm? e) Given that there are 5 calls in the first 20 minutes, what is the probability that all 5 calls are within the first 10 minutes? f) What is the expected total number of calls received within 2 consecutive 4-hour shifts