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Unformatted text preview: Chapter Three Chapter Three Discrete Random Variables & Probability Distributions Random Variable Random Variable A “rule” that assigns a number to each outcome in the sample space. Types of RVs Types of RVs Discrete: Possible values either constitute a finite set or else an infinite sequence in which there is a first element, second element, etc. Continuous: Possible values consists of an entire interval on the number line. Bernoulli RV Bernoulli RV A RV with only two possible values: 0 or 1 Probability Distribution Probability Distribution A mathematical model that relates the value of a RV with the probability of occurrence of that value in the population. Probability Measures Probability Measures Given real numbers r1 & r2: Probability that a RV a) Equals r1 b) Is greater than r1 c) Is between r1 & r2 d) Is less than r1 e) Is less than or equal to r1 Example: Discrete Probability Distribution Example: Discrete Probability Distribution A machine produces 3 items per day. QC inspection assigns to each item at the end of the day: defective or non defective. Assume that each point in the sample space has equal probability. If RV X is the number of defective units at the end of the day, what is the probability distribution for X? Example: Discrete Probability Distribution Example: Discrete Probability Distribution Continued from Previous Example: If a nondefective items yields a profit of $1,000 whereas a defective item results in a loss of $250. What is the probability distribution for the total profit for a day? Discrete Probability Distribution Discrete Probability Distribution A mfg. plant has 3 student & 3 veteran engineers assigned to the shop floor. Two engineers are chosen at random for a special project. Let the RV X denote the number of student engineers selected. Find the probability distribution for X. Discrete Probability Distribution Discrete Probability Distribution Starting at a fixed time, we observe the make of each car passing by a certain point until a Ford passes by. Let p = P(Ford) & RV X defined as the number of cars observed. Find the probability distribution for X. Example: PMF Example: PMF Consider the number of cells exposed to antigencarrying lymphocytes in the presence of polyethylene glycol to obtain first fusion. The probability that a given cell will fuse is known to be ½. Assuming that the cells behave independently, find the probability distribution for the number of cells required for first fusion. What is the probability that four or more cells require exposure to obtain the first fusion? F(x) = P(X<=x) = ∑ p(y) F(x) = P(X<=x) = OR y: y<=x P(a<=X<=b) = F(b) – F(a ) Example CDF Example CDF Let X denote the RV which is the toss of a loaded die. The probability distribution of X follows: p(1) = p(2) = 1/6 p(3) = 1/12 p(4) = p(5) = ¼ p(6) = x a) Find the value of x. b) Evaluate the CDF at 3.6. c) Find p(3<=X<=5). CDF to Find Probabilities CDF to Find Probabilities A mailorder business has 6 telephones. Let RV X denote the number of phones in use at a specified time. The pmf of X is given as follows: x | 0 1 2 3 4 5 6 p(x) | .01 .03 .13 .25 .39 .17 .02 What is the probability of: a) at most 3 lines are in use? b) at least 5 lines are in use? c) between 2 & 4 lines, inclusive are in use? Example of Expect Values Example of Expect Values Whatis the expected value of the RV X where X is the value on the face of a die? Expected Value Expected Value
What is the expected value for the RV X, which is the sum of the upturned faces when two dice are tossed? Expected Value Expected Value A university has 15,000 students. Let RV X equal the number of courses for which a randomly selected student is registered. The pmf of X follows: x | 1 2 3 4 5 6 7 p(x) | .01 .03 .13 .25 .39 .17 .02 Find the expected value of X. Expected Value of Bernoulli RV Expected Value of Bernoulli RV pmf: p(x) = 1 – p for x = 0 p for x = 1 What is the expected value of X? Expected Value of a Function Expected Value of a Function E[h(X)] = ∑ h(k)p(k) All k RV X has set of possible values k & pmf p(x). V(X) = σ = ∑ (k µ ) p(k) All k RV X has a set of possible values k with pmf p(x) and expected value µ . Example of Variance & E(X) Example of Variance & E(X) A discrete pmf is given by: p(x) = Ax x = 0,1,2,3,4,5 Determine A. What is the probability that x<=3? What is the expected value of X? What is the Variance & SD? Example of E(X) & Variance Example of E(X) & Variance Given the following pmf for RV X: x p(x) 0 1/8 1¼ 2 3/8 3¼ Find the E(X). Find the Variance. (Use Short Cut) Expect Value & Variance Expect Value & Variance Three engineering students volunteer for a taste test to compare Coke & Pepsi. Each student samples 2 identical looking cups & decides which beverage he or she prefers. How many students do we expect to pick Pepsi; knowing that 3/5 of all students prefer Pepsi over Coke? b) Find the Variance of the RV. Discrete Probability Distributions Discrete Probability Distributions Binomial Negative Binomial Hypergeometric Poisson Binomial Probability Distribution Binomial Probability Distribution Binomial Experiment: > Consists of a sequence of n trials, where n is fixed in advance. > The trials are identical & each trial can result in one of the same two possible outcomes. > The trials are independent. > The probability of the outcomes is constant & is equal to p & 1p. Example Discrete pmf Example Discrete pmf You draw at random a 20 piece sample from a group of 300 parts in storage where 10% of the parts are known to be out of specification. What is the probability that 1 part in your sample will be out of spec.? Binomial Example Binomial Example A coin is tossed 4 times. What is the pmf for the RV X; the number heads? What is the probability of having 3 or fewer Heads? Binomial Example Binomial Example There are 5 intermittent loads connected to a power supply. Each load demands either 2w or no power. The probability of demanding 2w is ¼ for each load. The demands are independent. What is the pmf for the RV X, the power required? Example Binomial pmf Example Binomial pmf A lot of 300 manufactured baseballs contains 5% defects. If a sample of 5 baseballs is tested, what is the probability of discovering at least one defect. Negative Binomial pmf Negative Binomial pmf Experiment: > The trials are independent. > Each trial can result in either a success (S) or a failure (F). > The probability of the outcomes is constant from trial to trial. > The experiment continues until a total of r successeshave been observed, where r is a specified positive integer of interest. Negative Binomial Example Negative Binomial Example An engineering manager needs to recruit 5 graduating student engineers. Let p = P (a randomly selected student agrees to be hired). If p = 0.20, what is the probability that 15 student engineers must be given an offer before 5 are found who accept? Discrete pmf Example Discrete pmf Example Your oil exploration crew is testing for well sites. Historically, the probability of finding oil in your present geographical location is 1/20. HQs needs your crew to locate 2 oil producing wells within 2 weeks. If setup & testing for oil takes 1 day, what is the probability that it will take less than 2 working weeks to find these 2 spots? nb Probability Distribution nb Probability Distribution You have passed your ISE 261 at the end of the semester & decide to celebrate. For whatever reason, you are arrested for a misdemeanor & sentenced for 90 days in the county jail. The judge being a student of Probability Theory decides to give you an option. You can have the full 90 days or you can elect to leave jail after rolling 1 die for 16 straight even numbers. Which option do you decide to take? Remember, the judge will only give you a guard for affirming your rolls for 8 hours per day. Your ability to roll one die for 16 rolls is 2 minutes & the judge insists on blocks of 16 rolls. Geometric Probability Distribution Geometric Probability Distribution You have conducted a series of experiments to reduce the proportion of scrapped battery cells to 1% in your manufacturing plant. Now, what is the probability of testing 51 cells without finding a defect until the last cell? Hypergeometric pmf Hypergeometric pmf Experiment: > Consists of N individuals, objects, or elements (a finite population). > Each individual can be characterized as a success (S) or failure (F), & there a M successes in the population. > A sample of n individuals is selected without replacement in such a way that each subset of size n is equally likely to be chosen. h(x; n, M, N) = C C Example Hypergeometric pmf Example Hypergeometric pmf From a group of 20 EE students, you select 10 for employment. What is the probability that the 10 selected include all the 5 best engineers in the group of 20? Hypergeometric Example Hypergeometric Example Your manufactured product is shipped in lots of 20. Testing is costly, so you sample production rather than use 100% inspection. A sampling plan constructed to minimize the number of defectives shipped to customers calls for sampling 5 items from each lot & rejecting the lot if more than 1 defective is observed. If a lot contains 4 defectives, what is the probability that it will be rejected? Binomial Approximation Binomial Approximation In any binomial experiment in which n is large & p is small, the binomial pmf is approximately equal to the poisson pmf where λ = np. Rule of Thumb n => 100; p <= .01; np <= 20 Mean & Variance of Poisson Mean & Variance of Poisson E(X) = V(X) = λ Poisson pmf Example Poisson pmf Example You are in charge of a PCB operation. It is known that the distribution of the number of solder balls occurring on 1 board in this process is Poisson with λ = 1.0. You have 1,000 PCBs in this process & would like to estimate how many boards would have solder balls on them. Poisson Probability Distribution Poisson Probability Distribution A radioactive substance emits alpha particles. The number of particles reaching a counter during an interval of 1 second has been observed to have a Poisson probability distribution with λ = 10. What is the probability that the RV X, the number of particles reaching the counter during 1 second is 3.0? Poisson pmf Poisson pmf A machine produces sheet metal where the RV X, the number of flaws per yard follows a Poisson distribution. The average number of flaws per yard is 2.0. Plot the pmf for RVX. Poisson Approximation Example Poisson Approximation Example Electrical resistors are packaged 200 to a continuous feed ribbon. Historically, 1.5 percent of the resistors manufactured by a machine are defective. Compute the pmf of the number of defective resistors on a ribbon and compare it to the Poisson approximation. (Stop at x = λ ) Poisson Process Example Poisson Process Example As system’s engineering manager, you have devised a random system of police patrol so that an officer may only visit a given location in his area with the Poisson RV X = 0,1,2,3,… times per 1hour period. The system is arranged so that he visits each location on an average of once per hour. Calculate the probability that an officer will miss a given location during a halfhour period. What is the probability that an officer will visit once? Twice? At least once? Poisson pmf Poisson pmf In an industrial plant there is a dc power supply in continuous use. The known failure rate is λ = 0.40 per year & replacement supplies are delivered at 6 months intervals. If the probability of running out of replacement power supplies is to be limited to 0.01, how many replacement power supplies should the operations engineer have onhand at the beginning of the 6month interval? Poisson Process Poisson Process The number of hits on the ISE 261 Web site from Noon to 12:30 PM on the day before a Quiz follows a Poisson distribution. The mean rate is 3 per minute. Find the probability that there will be exactly 10 hits in the next 5 minutes. Let RV X be the number of hits in t minutes, what is the pmf in terms of t minutes? Distribution Parameters Distribution Parameters Binomial n & p Negative Binomial r & p Hypergeometic n, M, & N Poisson λ Hypergeometric: nM nM(NM)(N n) N N (N1) 2 ...
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This note was uploaded on 06/10/2009 for the course ISE 261 taught by Professor Koon during the Spring '08 term at Binghamton University.
- Spring '08