Probability Models for Engineers
Probability Models for Engineers STAT 3345
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Date Created: 09/17/15
Discrete Random Variables and Probability Distributions Random Variables and Probability Distributions 0 A random variable is a function that assigns a numerical value to each sample point in a sample space A random variable re ects the aspect of a random experiment that is of interest to us There are two types of random variables 1 Discrete random variable 2 Continuous random variable A random variable is discrete if it can assume only a countable number of values A random variable is continuous if it can assume an uncountable number of values Discrete Probability Distribution 0 A table formula or graph that lists all possible values a discrete random variable can assume together with associated probabilities is called a discrete probability distribution 0 To calculate PX x the probability that the random variable X assumes the value x add the probabilities of all the simple events for which X is equal to x Example 1 Find the probability distribution of the random variableX describing the number of heads that turnup when a coin is ipped twice Solution The possible values are 0 1 and 2 PX0PTT14 PX 1 PTHPHT 12 PX2PHH14 Requirements of discrete probability distribution If a random variable can take values x1 then the following must be true 10 S pxl S 1 for all x 2 2 m 1 all x The probability distribution can be used to calculate probabilities of different events Probabilities as relative frequencies In practice often probabilities are estimated from relative frequencies Example 2 The numbers of cars a dealer is selling daily were recorded in the last 100 days This data was summarized as follows Daily sales Freguency 5 l 15 2 35 3 25 4 E 100 1 Estimate the probability distribution 2 State the probability of selling more than 2 cars a day Solution 1 The estimated probability distribution table x 0 1 2 3 4 PX x 05 15 35 25 20 2 Pmore than 2 cars a day PX gt 2 PX 3 PX 4 25 20 45 Joint Distribution Consider two discrete random variables X that takes values x x 1 2xn Y that takes values y1 y2 ym If need to see the relation ship between these two random variables the distributions of X and Y separately are not going to provide the story For this we need the joint distribution table X x1 x2 x1 xn PY y Y yl p11 p12 p11 pln p10 yZ p21 p22 p21 pin p20 y iv1 P12 P p p ym PM pm 1 Pm Pm PX x iv1 172 p p 1 In this table 0 Goint probability p PY y and X x1 0 marginal probability of X p p11 p21 pm PX x1 0 marginal probability of Y p p1 pj2 pjn PY yj Example 2a A fair coin is ipped three times LetXbe a number of heads Y is equal to 1 if the rst ip is head and it is 0 if it is tail Give the joint distribution of X and Y Solution X 0 l 2 3 PY y Y 0 18 28 18 0 12 1 0 18 28 18 12 PX x 18 38 38 18 1 The expected value Given a discrete random variable X with values x that occur with probabilities pxl the expected value of X is EX 296110061 The expected value of a random variable X is the weighted average of the possible values it can assume where the weights are the corresponding probabilities of each Xi Laws of Expected Value 0 Ec c o EcX cEX o EXYEXEY o E X Y E X E Y if X and Y are independent random variables Note X and Y are said to be independent if for any possible values x1 and y j of X and Y respectively we have PX xlY yPX xlPY yj ie in terms of the joint distribution P P XPW Variance Let X be a discrete random variable with possible values x that occur with probabilities px and let EX u The variance of X is de ned to be VarX 02 EX 2 Z x ml pee The variance is the weighted average of the squared deviations of the values of X from their mean u where the weights are the corresponding probabilities of each x1 Properties of the variance 0 Varconst 0 VaraX a2 VarX VarXY VarX VarY2COVX Y where COVX Y EX EXY EY If X and Y are independent then C O VX Y 0 and VarX Y VarX VarY If VarX 0 then X const Standard deviation The standard deviation of a random variable X denoted 039 is the positive square root of the variance of X Example 3 The total number of cars to be sold next week is described by the following probability distribution x 0 1 2 3 4 px 05 15 35 25 20 Determine the expected value and standard deviation of random variable X the number ofcars sold Solution EX0gtlt051gtlt152gtlt353gtlt254gtlt2024 VarX 0 242 X 05 1 242 X 15 2 242 X 35 3 242 X 25 4 242 X 2 124 039 lVarX V124 1114 An expected value of f X is EfX Zfx Px Bernoulli trial The Bernoulli trial can result in only one out of two outcomes Typical cases where the Bernoulli trial applies 0 A coin ipped results in heads or tails 0 An election candidate wins or loses 0 An employee is male or female 0 A car uses 87 octane gasoline or another gasoline Binomial experiment There are n Bernoulli trials 71 is nite and xed Each trial can result in a success or a failure The probability p of success is the same for all the trials All the trials of the experiment are independent bP NE Binomial Random Variable The binomial random variable counts the number of successes in n trials of the binomial experiment By de nition this is a discrete random variable Calculating the Binomial Probability In general the binomial probability is calculated by PX x 1796 CEPXU P where n n xn x is the number of different ways of choosing x objects from a total of 71 objects Here 71 l X 2 X 3 X X n by convention 0 l Example 4 Suppose that we have a group of 4 people say A B C and D How many different pairs can we select from this group Solution The answer is 4 choose 2 4 4 lgtlt2gtlt3gtlt4 2 2gtlt2 1gtlt2gtlt1gtlt2 Indeed we have six pairs AB AC AD BC BD and CD Mean and variance of binomial random variable E X np VaVX quot1170 P Poisson Distribution The Poisson experiment typically ts cases of rare events that occur over a xed amount of time or within a speci ed region Typical cases The number of errors a typist makes per page The number of customers entering a service station per hour The number of telephone calls received by a switchboard per hour Poisson Experiment Properties of the Poisson experiment 0 The number of successes events that occur in a certain time interval is independent of the number of successes that occur in another nonoverlapping time interval 0 The average number of a success in a certain time interval is the same for all time intervals of the same size proportional to the length of the interval 0 The probability that two or more successes will occur in an interval approaches zero as the interval becomes smaller The Poisson Random Variable The Poisson variable indicates the number of successes that occur during a given time interval or in a speci c region in a Poisson experiment Probability Distribution of the Poisson Random Variable e39 u x EXVarXu PX x pm x 012 Poisson Approximation of the Binomial When n is very large binomial probability table may not be available If p is very small p lt 05 we can approximate the binomial probabilities using the Poisson distribution More speci cally we have the following approximation PX x E PXPoisson np x where the binomial distribution with parameters 71 and p and Poisson with rip Binomial pm The Geometric Random Variable Let X be the number of the trial on which the first success occurs in a binomial experiment PX x 1 PHP EX1p VaVX 11101172 The Hypergeometric Random Variable Suppose that a population contains a nite number N of elements that possess one of two characteristics Thus r of the elements might be red and b N r black A sample of n elements is randomly selected from the population and the random variable of interest is X the number of red elements in the sample This random variable has what is known as the hypergeometric probability distribution C r x CNrr NquotEXnrNVarXn N VN quot PXx C W N N l Exercises 1 Ten thousand Instant Money lottery tickets were sold One ticket has a face value of 1000 5 tickets have a face value of 500 each 20 tickets are worth 100 each 500 are worth 1 each and the rest are losers Let X face value of aticket that you buy Find the probability distribution for X N An altered die has one dot on one face two dots on three faces and three dots on two faces The die is to be tossed once Let X be the number of dots on the upturned face Find the mean and variance of X 3 A card is to be selected from an ordinary deck of 52 cards Suppose that a casino will pay 10 if you select an ace If you fail to select an ace you are required to pay the casino 1 a If you play this game once how much money does the 4 V39 05 gt1 9 0 casino expect to win b If you play the game 26 times how much money does the casino expect to win A bus company is interested in two potential contracts one for an express and the other for local stops The probabilities that the bids will be accepted are 70 and 50 with costs of 500 and 750 respectively The estimated total incomes are 6000 and 10000 respectively If the company is allowed only one bid which bid should it enter In the game of craps a player rolls two dice If the first roll results in a sum of 7 or 11 the player wins If the first roll results in a 2 3 or 12 the player loses If the sum on the first roll is 4 5 6 8 9 or 10 the player keeps rolling until he throws a 7 or the original value If the outcome is a 7 the player loses If it is the original value the player wins The probability that a player will win is 493 Suppose that a player pays 5 to a casino if he loses and is paid 4 for a win What is the expected loss for the player if he plays a one game b ten games A high school class decides to raise some money by conducting a raf e The students plan to sell 2000 tickets at 1 apiece They will give one prize of 100 two prizes of 50 and three prizes of 25 If you plan to purchase one ticket what are your expected net winnings Hint The probability of getting the 100 ticket is 12000 of getting a 50 ticket is 22000 and of getting a 25 ticket is 32000 Forty percent of the students at a large university are in favor of a ban on drinking in the dormitories Suppose 15 students are to be randomly selected Find the probability that a Seven favor the ban b Fewer than 4 favor the ban c More than 2 favor the ban Sixty percent of the voters in a large town are opposed to a proposed development If 20 voters are selected at random find the probability that a Ten are opposed to the proposed development b More than 13 are opposed to the proposed development c Fewer than 10 are opposed to the proposed development Of a population of consumers 60 are reputed to prefer a particular brand A of toothpaste If a group of randomly selected consumers is interviewed what is the probability that exactly five people have to be interviewed to encounter the first consumer who prefers brand A At least five people In responding to a survey question on a sensitive topic such as quotHave you ever tried marijuanaquot many people prefer not to respond in the affirmative Suppose that 80 the population have not tried marijuana and all of those individuals will truthfully answer to your question The remaining 20 of the population have tried marijuana and 70 of those individuals will lie Derive the probability distribution of the number of people you would need to question in order to obtain a single affirmative response 11 Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour During a given hour what are the probabilities that No more than three customers arrive At least two customers arrive Exactly ve customers arrive If it takes approximately ten minutes to serve each customer nd the mean and variance of the total service time for customers arriving during a 1hour period e Is it likely that the total service time will exceed 25 hours f What is the probability that exactly two customers arrive in the twohour period oftime between a 200 PM and 400 PM one continuous twohour period b l00 PM and 200 PM or between 300 PM and 400 PM two separate onehour periods that total two hours 999 12 The number of typing errors made by a typist hag a Poisson distribution with an average of four errors per page If more than four errors appear on a given page the typist must retype the whole page What is the probability that a randomly selected page does not need to be retyped A shipment of 20 cameras includes 3 that are defective What is the minimum number of cameras that must be selected I fwe require that Pat least 1 defective gt 8 Seed are often treated with fungicides to protect them in poor draining wet environments A smallscale trial involving live treated and live untreated seeds was conducted prior to a largescale experiment to explore how much fungicide to apply The seeds were planted in wet soil and the number of emerging plants was counted If the solution was not effective and four plants actually sprouted what is the probability that a All four plants emerged from treated seeds b Three or fewer emerged from treated seeds c At least one emerged from untreated seeds Multivariate Probability Distributions Joint Discrete Distribution Let X1 and X 2 be discrete random variable The joint probability distribution of X1 and X 2 is given by px1x2PX1 x15X2 x2 de ned for all real numbers x1 and x2 The function px1x2 is called the joint probabilityfunction of X1 and X2 The marginal probabilityfunctions of X1 and X2 respectively are given by p1x1zpx1 x2 and p2x2zpx1 x2 Joint Continuous Distribution Let X1 and X 2 be continuous random variable The joint probability density function of X1 and X 2 is a nonnegative function f x1 x2 such that 171172 Pa1 le 131612 3X2 132 ijx1x2dxldx2 The marginal probability density functions of X1 and X 2 respectively are given by f on jfltx1x2gtdx2 and M jfltx1 gtde The conditional probability density function of X1 given X 2 x2 is given by fxlxZ fxlxZ fzm 0 otherwise iffzxZgt0 and the conditional probability density function of X 2 given X1 x1 is given by roux 79 1 1 0 fx2x1 mg quotmp 0 otherwise Independence Discrete random variables X1 and X 2 are said to be independent if 100999 11710601172 xi for all real x1 and x2 Continuous random variables X1 and X 2 are said to be independent if fx1x2 f1xlfzxz for all real x1 and x2 Expected value of function of random variables Let X1 and X 2 be two random variables and gx1 x2 is a function of x1 x2 Then 2 Z gx1 x2 px1 x2 in case of discrete random variables EgX1X2 no no I J gx1 x2 f x1 x2 in case of continuous random variables Covariance and correlation The covariance between two random variable X1 and X 2 is given by C0VX1X2 EX1 EX1X2 EX2 EX1X2EX1EX2 The correlation coef cient is given by COVX1X2 p VarX1VarX2 39 The correlation coefficient is always between 1 and l The sign of covariance gives direction of the association and correlation measures the strength of the association Conditional expectation The conditional expectation of X1 given X 2 x2 is given by 2x1 px1 l x2 in case of discrete random variables 1C1 EX1lX2x2 0 1x1 f x1 l x2 in case of continuous randomvariables Exercises With X1 denoting the amount of gasoline stocked in a bulk tank at the beginning of a week and X 2 the amount sold during the week Y XI X 2 represents the amount left over at the end of the week The joint density function of X 1 X2 is given by 3x1if0Sx2Sx1Sl 0 otherwise fx1x2 Find marginal pdfs of X1 and X 2 Find the means and variances of X1 and X 2 Find the covariance of X1 and X 2 bP N Find the mean and variance of Y Probability Discrete Sample Space Probability space A random experiment is a process or course of action whose outcome is uncertain A sample space S of a random experiment is a set of all possible outcomes simple events of the experiment The outcomes must be mutually exclusive and exhaustive An event is a subset of the sample space that is any collection of one or more outcomes Suppose S is a sample space associated with an experiment To every event A in S A is a subset of S we assign a number PA called the probability of A so that the following axioms hold Axiom 1 PA Z 0 Axiom 2 PS 1 Axiom 3 If A1A2A3 form a sequence of pairwise mutually exclusive events in S that is A MA ifi jthen PA1 UA2 uA3 U ZPAI Notice that the de nition states only the conditions an assignment of probabilities must satisfy it does not tell us how to assign speci c probabilities to events Assignment of Probabilities Given a sample finite spaceSE1E2En the following properties for the probability PEl of outcomes E1 must hold 1 0SPltE1S1 for eachi 2 PE11 11 If event E consists of finite number of outcomes st u 2 Then PE PSPtPuPz Probability Rules Complement rule PA 1 PA Inclusion Exclusion principle PA U B PAPB PA n B Experiments with equally likely outcomes Suppose that A is an event in a nite sample space S of N equally likely outcomes Then Number ofoutcomes in A PA N Conditional Probability o The probability of an event when partial knowledge about the outcome of an experiment is known is called conditional probability 0 We use the notation PA B for the conditional probability that event A occurs given that event B has occurred PA r B P A B w 133 0 In case of experiments with equally likely outcomes Number of outcomes in A r B P A B i Number of outcomes in B Independent Events Events A and B are independent iff PA n B PA PrB If two events A and B are independent then PA B PA and PB A PB Product Rule For any two events A and B PA n B PAPB A PBPA B Bayes Theorem If 313 B are mutually exclusive events and if S B1 U B2 U U B then for any 2 event A in S PBl39PA Bl PBl39PA BlPBz39PA BzPBn39PA Bi 13031 A PBz39PAiBz PBl39PA BlPBz39PA BzPBnPA Bi PBz A and so forth Note that PAPBl39PAiBlPBz39PABzPBn39PABn Exercises 1 N E 4 Balls in an Urn An urn contains ve red balls and four white balls A sample of two balls is selected at random from the urn What is the probability that at least one of the balls is red Coin Tosses A coin is to be tossed seven times What is the probability of obtaining four heads and three tails Opinion Polling Of the 15 members on a Senate committee 10 plan to vote quotyesquot and 5 plan to vote quotnoquot on an important issue A reporter attempts to predict the outcome of the vote by questioning siX of the senators Find the probability that this sample is precisely representative of the nal vote That is nd the probability that four of the siX senators questioned plan to vote quotyesquot Boys and Girls A couple decides to have four children What is the probability that they will have more girls than boys 5 0 gt1 9 Coin Tosses A coin is to be tossed siX times What is the probability of obtaining exactly three heads Selecting Students In a classroom of children 12 boys and 10 girls in which seven students are chosen to go to the blackboard a What is the probability that at least two girls are chosen b What is the probability that no boys are chosen c What is the probability that the rst three children chosen are boys Presidential Choices There were 16 presidents of the Continental Congresses from 1774 to 1788 Each of the ve students in a seminar in American history chooses one of these on which to do a report If all presidents are equally likely to be chosen calculate the probability that at least two students choose the same president Curriculum change A change was proposed in the mathematics curriculum at a college The mathematics majors were asked whether they approved of the proposed change The results of the survey follow Approved No opinion Did not approve Female 21 12 Male 14 10 7 Suppose that a mathematics major is selected by chance Find the probability that a The student is male and approves of the proposed change b The student is male c The student approves of the proposed change given the student is male d The student is male given the student does not approve of the proposed change e The student is female given no opinion