Introduction To Probability Models
Introduction To Probability Models STAT 22500
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Date Created: 09/19/15
Stat 225 7 Final Exam Review Spring 2009 Solving the following problems will be good practice for the upcoming exam Try them on your own over the next couple of daysisolutions will be posted shortly 1 F 9 For each random variable X below7 name its distribution and parameters If an approximate distribution is appropriate7 give this as well On election day7 voters arrive at a polling station according to a Poisson process with an average rate of 150 per hour In this precinct7 suppose 55 of the registered voters vote Republican and 45 vote Democrat ignore all other parties Suppose that the new electronic voting machines correctly count a vote with probability 095 a X the number of votes cast during the rst three hours after the polling station opens X the number of correctly counted votes among the rst 100 votes cast X the number of times that two voters cast their votes within 20 seconds of eachother among the rst 20 voters X the time between the next two Democrat votes X the number of votes correctly counted7 assuming all 5000 registered voters cast their votes X the time that the local candidate for state representative will cast her vote7 assuming she will vote sometime between 900 am and 1200 pm X the number of votes until someone votes Republican and the vote is not correctly counted A continuous random variable X has CDF given by 07 lt0 FXx if 0 zlt2 l7 zgt2 a b C d Compute the probability that X is between 0 and 05 Find the 45th percentile of X Find the PDF of X Find the mean and standard deviation of X Cars cross a certain point in the highway in accordance with a Poisson process with rate A 3 cars per minute Suppose it takes 10 seconds for Al to run across this particular street a If Al blindly runs across the highway7 then what is the probability that he will be uninjured Assume that if he is on the highway when a car passes by7 then he will be injured U 03 0 Suppose now that Al is agile enough to dodge a single car but if he encounters two or more cars while attempting to cross the road7 then he is injured What is the probability that he will be unhurt The PDF of a continuous random variable X is given in the following formula 5 m A A A 1 lt z lt 2 fXx 12 67 A 7 07 otherw1se Verify that this is a legitimate PDF Find the expected value of X Find the CDF What is the probability that X is equal to 1 What is the probability that X is greater than 17 given X is greater than 0 What is the 80th percentile of X If four independent observations of X are taken7 each with PDF given above7 what is the probability that at least three of them are greater than 0 If independent observations of X are taken7 each with PDF given above7 how many observations would you expect to be needed until you see your 1st ob servation that is greater than 1 A continuous random variable X has PDF fx and CDF Decide if each of the following statements are TRUE or FALSEFno proofs required A D can have the value 25 f can have the value 25 AA 0 lt7 The graph of can have ups and downs A CL The graph of f can have ups and downs The integral of is The integral of f is The derivative of is f The derivative of f is can be negative AA A 37qu 1 f can be negative A Wu The area under the curve is 1 A The area under the curve f is 1 f3 is the same as PX 3 F3 is the same as PX S 3 m n Jennifer7s house is located near some railroad tracks She can hear an average of 4 trains go by her house per hour I 00 Q a Jennifer enjoys spending her afternoons outdoors If an afternoon consists of four hours from noon to 400 what is the probability that she hears between 14 and 16 trains go by b If she spends every afternoon outside how many afternoons should she expect to wait until exactly 14 trains go by in an afternoon c Jennifer is waiting until she hears the next train go by Given that she has waited 15 minutes what is the probability she will not have to wait longer than 30 minutes d In one afternoon she heard exactly 18 trains go by What is the probability she heard at least 2 trains in the rst hour The temperature in F on June 1st in Chicago is Norrnally distributed with mean 76 F and standard deviation of 3 F a Find and interpret the 75th percentile b What is the probability that the June 1st temperature will exceed 80 F c To investigate the occurance of global warming rneteorologists will record the temperature on June 1st over the next 3 years Assume the June 1st temperatures in consecutive years are independent What is the probability that the June 1st ternperature exceeds 80 F in each of the next 3 years d Suppose instead of F the temperature in C is desired If x is a temperature in F then y 32 7 32 is the corresponding temperature in C Give the distribution and pararneters of the June 1st temperature in C Scientists are analyzing data obtained from a road test of cars with Brand T tires They nd that the stopping time for a speci c speed and a speci c distance follows a Normal distribution with mean 16 seconds and standard deviation 025 seconds a What is the probability that a car will stop in less than 09 seconds b What is the probability that a car will stop between 1 and 2 seconds c Given that a car takes at least 2 seconds to stop what is the probability it will take less than 24 seconds to stop Consider the grid of points shown below Suppose that starting at the point labeled A you can go one step up or one step to the right at each move This is continued until the point B is reached a How many different paths from A to B are possible Hint You need to make 4 steps to the right and 3 steps up b Find the number of paths from point A to point B which go through point C H O H H HHH JgtLON A 1 Drew enjoys eating fastifoodi The number of hamburgers Drew eats in a weeks time7 say X7 is a discrete random variable with PMF given in the following table I 0 1 2 3 pXz 0135 01275 01225 011 0105 a Let Y denote the average number of hamburgers Drew eats per week over a years time 52 weeks What is the distribution of Y7 Give the parameters of this distribution and explain why this distribution can be used b What is the probability that Y is between 1 and 1157 inclusive c What is the probability that in each of the next seven years7 Drew7s weekly hamburger average is between 1 and 1157 d Let 5 denote the total number of hamburgers Drew eats in 250 weeks Give the distribution of S and its parametersi e What is the probability that S is between 285 and 3207 inclusive 1 The number of runs scored per game by a particular major league baseball team is a discrete random variable with mean 5 runs and standard deviation 3 runs A baseball season consists of 182 games let X denote the average number of runs per game for this particular team over the 1827game season a Give an approximate distribution of X with parameters and explain why this approximation is valid b Find and interpret the 90th percentile of this distribution c What is the probability that X exceeds 514 runsi7 1 Suppose X N NW 602 4 Find EX2i i If X N Unif01 calculate EM 1 Suppose twelve percent of a population is lefthandedi Let S be the number of leftihanders in a highischool graduating class of 200 students Approximate the probability PS g 207 and state your assumptions 4 15 H 1 English and American spellings are rigour and rigor respectively A man staying at a Parisian hotel writes this word7 and a letter taken at random from his spelling is found to be a vowel If 40 percent of the English speaking men at the hotel are English and 60 percent are Americans7 what is the probability that the writer is English You and a friend go out to dinner and7 to decide who pays7 you each roll a four sided die with the following probabilities x l 1 2 3 4 Pdie lands on m l 025 025 035 015 Whoever rolls the lowest number must pay for dinneriif there7s a tie7 you each pay for your own dinner Find the probability you get a free dinner Hint Use the total probability formula Suppose you have two coins These coins look and feel identical7 but coin A lands on tails with probability 0005 and coin B lands on tails with probability 002 De ne two random variables X number of tails in 400 independent ips of coin A Y number of tails in 400 independent ips of coin B a Give the exact distribution for X and its parameters b Give an approximate distribution for X and its parameters you chose this approximation Explain why A O V Use your answer to b to nd the approximate probability of getting at least 3 tails in 400 ips of coin A A CL Give the exact distribution for Y and its parameters A D V Give an approximate distribution for Y and its parametersithis approx imation will not be the same as in part a Explain why you chose this approximation A l h V Use your answer to e to approximate probability of getting at least 3 tails in 400 ips of coin B Suppose both coins A and B are put into a bag Suppose you reach into the bag and pick a coin at random You can7t tell whether you picked coin A or coin B just by looking at it7 but then you ip the coin 400 times and there are at least 3 tails Given that there are at least 3 tails7 are you more likely to have chosen coin A or coin B Justify your answer using probability The number of hours X spent on homework each week by a college student can be modeled by a continuous random variable with the following PDF 704 if z gt 1 0 otherwise p O H 3 Find the value of c that makes this a valid PDF Find EX and VarX Suppose we take a sample of 300 randomly chosen college students What is the approximate probability that the total time spent studying by the 300 students is between 435 and 465 hours AAA O 793 VVV Researchers want to learn more about how newborn babies7 weight changes over the rst year of the life They take a random sample of 150 newborn babies7 and measure their weight at birth and then again at their one year birthday In the end7 the researchers found that the average weight increase for these 150 babies was 14 pounds with a standard deviation of 6 pounds a Construct a 99 con dence interval for the mean amount of weight increase for newborn babies in the rst year of life b Another research institute claims that the mean weight gain during a new born7s rst year is 20 pounds Based on your con dence interval in part a7 do you agree with this statement Explain why or why not A microchip manufacturer7s entire output is produced by three machines Machines A7 B7 and C produce 357 257 and 45 of the manufacturers microchips and 107 157 and 12 of their respective outputs are defective A quality control engineer randomly selects a microchip from a recently produced batch a What is the probability that the microchip selected is defective and was pro duced by Machine B b Find the probability that the selected microchip is defective c uppose a e ec 1ve microc 1p 1s se ec e a is e pro a 11y 1 was S d f t h l t d Wh t th b b l t t produced by Machine A Suppose we have a coin assumed to have probability p of landing heads7 but we don7t know the value of p To determine a set of likely values for p7 we ip the coin 100 times and count the number of heads X observed a Give a model for X b If X 57 is observed7 use this information to estimate p c Assuming 02 027 construct a 95 con dence interval for p using the infor mation in b that X 57 d How many times would we need to ip the coin if we want the margin of error of the 95 Cl for p to be at most 005 Suppose A7 B and C are events such that PA B 0 gt 0 a Prove that PA B 0 PAPBlAPClA B b Prove that if A and B are independent7 then A and B0 are independent c Prove that PA U B S PA PB D 00 ls is possible for an event A to be independent of itself Explain Let X and Y be independent discrete random variables with X N Berp 05 and Y N Binn 210 05 De ne Z X Y Write out the PMF table for Z Torn and Jerry each have a biased coin When ipped7 Torn7s coin shows heads with probability 06 and Jerry7s coin shows heads with probability 07 a Suppose they play a game where7 in each round7 both players ip their coin these rounds continue until both coins land on the same face ie7 HH or TT What is the probability they play more than 2 rounds b If they ip their coins together 10 tirnes7 whats the probability that they see different faces each time Suppose X has an Exponential distribution with rate parameter A a What is the probability that X is greater than half its mean b If the rate A were doubled7 would the probability in a increase7 decrease7 or remain the same A bag contains 10 red rnarbles7 20 blue rnarbles7 and 30 green rnarbles Six rnarbles are randomly chosen from the bag What is the probability that 1 red7 2 blue and 3 green rnarbles are selected if the sampling is a without replacement b with replacement A door rnat rnanufacturer produces rnats which are designed to be 5 feet long by 3 feet wide7 but there is some variability in the dimensions of each individual rnat Suppose the production manager knows that the standard deviation of the length of each mat is 3 inches a What is the probability that the total length of 300 randomly sarnpled rnats exceeds 1508 feet b As in a7 ifT represents the total length in feet of the 300 randomly selected rnats7 nd P1498 S T S 1504 c Let A denote the average length in feet of the 300 randomly selected rnats Find PA S 499 d If we were to look at the average length in a sample of n rnats7 how large would we need to make 71 so that our 99 con dence interval is only 002 feet wide
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