Question

2.Calculating probabilities and quantiles with RUse R and your knowledge of the common probability densities we’ve talked about inclass to answer the following questions.(a) A representative from PSU’s athletic department randomly selects PSU studentsoutside the HUB to see if they attended the last home men’s basketball game. Letp, the probability she selects such a person, be equal to 0.001, and letXdenotethe number of people she must survey until she finds such a person.i. What is the probability the representative must select 4 people to find some-one who went to the game?ii. What is the probability that the representative must select more than 25people before finds one who attended the game?iii. How many people should we expect the representative to select before shefinds one who attended the last game? Also, what’s the variance?(b) A pack of wild Chihuahuas has a mean height of 7.5”, with a standard deviationof 1.5”.i. What proportion of these Chihuahuas are between 6” and 9” tall?ii. I want to know if a given Chihuahua is in the 90th percentile for height (inother words, is it taller than at least 90% of the other dogs?). For this to betrue, the Chihuahua must be at least how many inches tall?iii. What’s the probability that a dog from this pack is less than 3” tall?(c)SKIP THIS PROBLEM!Engineers designing the next generation of spaceshuttles plan to include two fuel pumps – one active, the other in reserve. If theprimary pump malfunctions, the second is automatically brought on line. Supposea typical mission is expected to require that fuel be pumped for at most 50 hours.According to the manufacturer’s specifications, pumps are expected to fail onceevery 100 hours. What are the chances that such a fuel pump system would notremain functioning for the full 50 hours?(d) Coliform bacteria are randomly distributed in a certain Arizona river at an averageconcentration of 1 bacterium per 20cc of water. Suppose we draw from the rivera test tube containing 10cc of water.i. What is the chance that the sample contains exactly 2 coliform bacteria?ii. What is the probability that the sample contains at least 4 coliform bacteria?iii. What is the expected number of bacteria we should find in 10cc of water?2 (e) You decide to take an elevator. Suppose the elevator doors are equally likely toopen between 0 and 120 seconds after you push the button to call the elevator.i. What is the expected time you have to wait to get on the elevator?ii. What is the probability the elevator doors open within 27 seconds?iii. Seventy-two percent of the time, the elevators doors will open within howmany seconds?(f) On average, a certain computer part lasts 10 years. LetXbe the length of timeuntil the part fails on your computer.i. What is the probability that the computer part lasts more than 7 years?ii. Eighty percent of the computer parts last at most how long?iii. What is the probability that your computer part lasts between 9 and 11 years?

Answer 1