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Chapter Three Discrete Random Variables & Probability Distributions

Chapter Three Discrete Random Variables & Probability Distributions. Random Variable A “rule” that assigns a number to each outcome in the sample space.

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Chapter Three Discrete Random Variables & Probability Distributions

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  1. Chapter ThreeDiscrete Random Variables & Probability Distributions

  2. Random VariableA “rule” that assigns a number to each outcome in the sample space.

  3. Types of RVsDiscrete: 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.

  4. Bernoulli RVA RV with only two possible values: 0 or 1

  5. Probability DistributionA mathematical model that relates the value of a RV with the probability of occurrence of that value in the population.

  6. Probability MeasuresGiven real numbers r1 & r2:Probability that a RVa) Equals r1b) Is greater than r1c) Is between r1 & r2d) Is less than r1e) Is less than or equal to r1

  7. Example: Discrete Probability DistributionA 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?

  8. Example: Discrete Probability DistributionContinued from Previous Example:If a non-defective 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?

  9. Discrete Probability DistributionA 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.

  10. Discrete Probability DistributionStarting 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.

  11. Example: PMFConsider the number of cells 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 first fusion?

  12. Cumulative Distribution FunctionCDF of a Discrete RV X with pmf p(x) is defined for every number x by:F(x) = P(X<=x) =  p(y) y: y<=xORF(n) = p(x1) + p(x2) + …+ p(xn)Where xn is the largest value of the x’s less than or equal to n.

  13. CDF PropertiesFor any two numbers a & b with a <= b, P(a<=X<=b) = F(b) – F(a-)Where a-represents the largest possible X value that is strictly less than a.If all integers for a, b, and values:P(a<= X<=b) = F(b) – F(a-1)Taking a=b: P(a) = F(a) – F(a-1)

  14. Example CDFLet 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) = 1/12p(4) = p(5) = ¼ p(6) = xa) Find the value of x.b) Evaluate the CDF at 3.6.c) Find p(3<=X<=5).

  15. CDF to Find ProbabilitiesA mail-order 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 6p(x) | .01 .03 .13 .25 .39 .17 .02What 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?

  16. Expected Values Of Discrete RVE(X) = X =  xnp(xn)All n 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.

  17. Example of Expect ValuesWhat is the expected value of the RV X where X is the value on the face of a die?

  18. Expected ValueWhat is the expected value for the RV X, which is the sum of the upturned faces when two dice are tossed?

  19. Expected ValueA 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 7p(x) | .01 .03 .13 .25 .39 .17 .02Find the expected value of X.

  20. Expected Value of Bernoulli RVpmf: p(x) = 1 – p for x = 0 p for x = 1What is the expected value of X?

  21. Expected Value of a FunctionE[h(X)] =  h(k)p(k)All kRV X has set of possible values k & pmf p(x).

  22. Variance of a Discrete RVV(X) = 2 =  (k - )2p(k)All kRV X has a set of possible values k with pmf p(x) and expected value .SD(X) = X = √(X2)

  23. Variance Shortcut MethodV(X)=[ k2p(k)] - 2 = E(X2)-[E(X)]2All kSteps: Find E(X2) Compute E(X) Square E(X) Subtract this value from E(X2)

  24. Example of Variance & E(X)A discrete pmf is given by: p(x) = Ax x = 0,1,2,3,4,5Determine A.What is the probability that x<=3? What is the expected value of X? What is the Variance & SD?

  25. Example of E(X) & VarianceGiven the following pmf for RV X:x p(x) 0 1/8 1 ¼ 2 3/8 3 ¼Find the E(X).Find the Variance. (Use Short Cut)

  26. Expect Value & VarianceThree 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.

  27. Discrete Probability DistributionsBinomial Negative Binomial Hypergeometric Poisson

  28. Binomial Probability DistributionBinomial 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 & 1-p.

  29. Binomial pmfb(x; n, p) = n! px (1-p)n-x x!(n-x)! x = 0,1,2,3,…,n

  30. Example Discrete pmfYou 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.?

  31. Binomial ExampleA 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?

  32. Binomial ExampleThere 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?

  33. Example Binomial pmfA 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.

  34. Negative Binomial pmfExperiment:> 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 successes have been observed, where r is a specified positive integer of interest.

  35. Negative Binomial pmfnb(x; r, p) = (x+r-1)! pr (1-p)x (r-1)!(x)! x = 0,1,2,3,…

  36. Negative Binomial ExampleAn 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?

  37. Discrete pmf ExampleYour 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?

  38. nb Probability DistributionYou 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.

  39. Geometric Probability DistributionYou 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?

  40. Hypergeometric pmfExperiment:> Consists of N individuals, objects, or elements (a finite population).> Each individual can be characterized as a success (S) or failure (F), & there aM 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.

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