Unformatted text preview: Types of RVs Types Discrete: Discrete: Possible values either constitute a finite set or else an infinite sequence in which there is a first element, second element, etc. second Continuous: Continuous: Possible values consists of an entire interval on the number line. line. Given real numbers r1 & Given r2: Probability that a RV a) Equals r1 b) Is greater than r1 b) c) Is between r1 & r2 c) d) Is less than r1 d) e) Is less than or equal e) Distribution Distribution A machine produces 3 items machine 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 Distribution Distribution Continued from Previous Continued Example: Example If a non-defective items If 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? profit Distribution Distribution A mfg. plant has 3 student mfg. & 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 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 Example: PMF Example: Consider the number of cells Consider exposed to antigen-carrying 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 Function Function CDF of a Discrete RV X with CDF pmf p(x) is defined for every number x by: number F(x) = P(X<=x) = ∑ p(y) F(x) p(y)
y: y<=x OR F(n) = p(x1) + p(x2) + …+ …+ p(xn) p(x Where xn is the largest value P(a<=X<=b) = F(b) – F(a ) F(a Where a represents the largest Where possible X value that is strictly less than a. less If all integers for a, b, and If values: values: P(a<= X<=b) = F(b) – F(a-1) F(a-1) Taking a=b: P(a) = F(a) – F(a- Example CDF Example Let X denote the RV which is Let the toss of a loaded die. The probability distribution of X follows: follows: p(1) = p(2) = 1/6 p(3) = p(3) 1/12 1/12 p(4) = p(5) = ¼ p(6) = x p(6) a) Find the value of x. b) Evaluate the CDF at 3.6. CDF to Find Probabilities CDF A mail-order business has 6 mail-order telephones. Let RV X denote the number of phones in use at a specified time. The pmf of X is given as follows: given x| 0123456 p(x) | .01 .03 .13 .25 .39 . 17 .02 What is the probability of: What a) at most 3 lines are in use? a) b) at least 5 lines are in use? c) between 2 & 4 lines, inclusive Discrete RV Discrete E(X) = µ E(X)
X All n = ∑ xnp(xn) Multiply every value that the RV can take on by the probability that it takes on this value; then add all of these terms together. these Example of Expect Values Values What is the expected What value of the RV X where X is the value on the face of a die? on Expected Value Expected
What is the expected What value for the RV X, which is the sum of the upturned faces when two dice are tossed? dice 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: pmf x |1 2 3 4 5 6 7 p(x) | .01 .03 .13 .25 .39 .17 .02 .02 Expected Value of Bernoulli RV Expected pmf: p(x) = 1 – p pmf: =0 p 1 for x for x = What is the expected What value of X? value Expected Value of a Function Function E[h(X)] = ∑ E[h(X)] h(k)p(k) h(k)p(k) All k All RV X has set of possible RV values k & pmf p(x). values Variance Shortcut Method Variance V(X)=[∑ k p(k)] - µ E(X2)-[E(X)]2 E(X
All k All 2 = Steps: Find E(X2) Find Compute E(X) Square E(X) Subtract this value from E(X2) A discrete pmf is given by: discrete p(x) = Ax x = p(x) 0,1,2,3,4,5 0,1,2,3,4,5
Determine A. Determine What is the probability that x<=3? x<=3? What is the expected value of X? X? Variance Variance Given the following pmf for Given RV X: RV x p(x) 0 1/8 1¼ 2 3/8 3¼ Find the E(X). Find the Variance. (Use Short Find 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? Coke? b) Find the Variance of the RV. Discrete Probability Distributions Distributions Binomial Negative Binomial Hypergeometric Poisson > Consists of a sequence of n trials, trials, where n is fixed in advance. advance. > The trials are identical & The each trial can result in one of the same same two possible outcomes. > The trials are independent. Binomial pmf Binomial b(x; n, p) = n! n-x (1-p) (1-p) x!(n-x)! x = 0,1,2,3,…,n p x Example Discrete pmf Example You draw at random a 20 You 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.? spec.? Binomial Example Binomial A coin is tossed 4 times. coin What is the pmf for the RV X; the number heads? RV What is the probability of What having 3 or fewer Heads? having Binomial Example Binomial There are 5 intermittent There 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? the Example Binomial pmf Example A lot of 300 lot manufactured baseballs contains 5% defects. If a sample of 5 baseballs is tested, what is the probability of discovering at least one defect. at Experiment: Experiment: > The trials are independent. > Each trial can result in either a success (S) or a failure (F). > The probability of the outcomes The is is constant from trial to trial. > The experiment continues until The a total of r successes have been observed, where r is a specified Negative Binomial pmf Negative nb(x; r, p) = (x+r-1)! p x (1-p) (1-p) (r-1)!(x)! r x = 0,1,2,3,… Negative Binomial Example Negative An engineering manager needs An 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? found 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 set-up & testing for oil takes 1 day, what is the probability that it will take less than 2 working weeks to find these 2 spots? spots? 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. for Which option do you decide to take? Which 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 Distribution Distribution You have conducted a series of You 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? defect objects, or objects, elements (a finite population). > Each individual can be Each characterized characterized as a success (S) or failure (F), & ), there a there M successes in the population. > A sample of n individuals is sample selected selected without replacement in such a way that way For RV X the number of S’s in the ’s sample. sample. h(x; n, M, N) = Cx,M Cnx,M nx,N-M Cn,N Integer x satisfies:
Max(0, n-N+M)<= x <= Min(n,M) Min(n,M) pmf pmf From a group of 20 EE From 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? of Hypergeometric Example Hypergeometric Your manufactured product is shipped Your 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? that Poisson Probability Distribution Poisson For RV X, the number of For random events that occur in a unit of time, space, or any other dimension often follows: other 0,1,2,… 0,1,2,… p(x; λ ) = e (λ )
-λ x x= x! x! λ >0 Binomial Approximation Binomial In any binomial experiment In 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 20 Mean & Variance of Poisson Poisson E(X) = V(X) = λ E(X) 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 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 What RV X, the number of particles Poisson pmf Poisson A machine produces sheet machine 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. 2.0. Example Example Electrical resistors are Electrical 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. Poisson Process Poisson Px(t) = e (α t) x!
-α t x For time interval t with parameter λ = α t. For 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 1-hour 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 halfwill hour period. What is the probability that an officer will What visit once? Twice? At least once? visit 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 on-hand at the beginning of the 6-month interval? interval? 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 Distribution Parameters Distribution Binomial Binomial Negative Binomial p Hypergeometic &N Poisson n&p r& n, M, λ Expected Values & Variances Expected pmf E(X) V(X) pmf E(X) V(X) Binomial: np np(1np p) Negative Binomial: Negative r(1-p) r(1-p) r(1-p) r(1-p) p p2 Hypergeometric: nM nM(NHypergeometric: nM nM(NM)(N-n) N N2 (N-1) (N-1) ...
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