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Chapter 5: Probability Distributions: Discrete Probability Distributions

Chapter 5: Probability Distributions: Discrete Probability Distributions. Learning Objectives. Identifying Types of Discrete Probability Distribution and their Respective Functional Representations.

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Chapter 5: Probability Distributions: Discrete Probability Distributions

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  1. Chapter 5: Probability Distributions:Discrete Probability Distributions

  2. Learning Objectives • Identifying Types of Discrete Probability Distribution and their Respective Functional Representations • Calculating the Mean and Variance of a Discrete Random Variable with each of the different discrete probability distribution identified.

  3. Random Variable

  4. Random Variables • Any variable that is used to represent the outcomes any experiment of interest to us is called Random Variable. • A random variable can assume (take) any value (Positive, negative, zero; finite, infinite; continuous and discrete values). • Depending upon the values it takes, we can identify two types of random variables: • Discrete Variables • Continuous Variables.

  5. Note that both types of variables can assume either a finite number of valuesor an infinite sequence of values.

  6. Example: JSL Appliances • Discrete random variable with a finite number of values Let x = number of TVs sold at a given store in one day. The number of TV units that can be sold in a given day is finite. It is also discrete: (0, 1, 2, 3, 4). X can be considered as Discrete Random Variable

  7. Example: JSL Appliances • Discrete random variable with an infinite sequence of values Let Y = number of customers arriving in a store in one day. Y can take on the values 0, 1, 2, . . . We can count the customers arriving. However, there is no finite upper limit on the number that might arrive.

  8. Random Variables Type Question Random Variable x Family size X = Number of dependents reported on tax return Discrete Continuous Y = Distance in miles from home to the store site Distance from home to a store Owning dog or cat Discrete Z = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)

  9. Probability Distributions

  10. Probability Distributions The probability distribution for a random variable is a distribution that describes the values that the random variable of interest takes. It is defined by a probability function, and is denoted by f(x) ---the probability of the values of the random variable.

  11. Probability Distributions For any probability function the following conditions must be satisfied: • f(x) > 0 2. f(x) = 1

  12. Discrete Probability Distributions

  13. Discrete Probability Distributions A Discrete Probability Distribution is a tabular, graphic or Functional representation of a Random Variable with discrete outcomes that follows the principle of probability distribution f(x) > 0 f(x) = 1

  14. Discrete Probability Distributions--Example • Using past data on sales, a tabular representation of the probability distribution for TV sales was developed. Number Units Soldof Days 0 80 1 50 2 40 3 10 4 20 200 xf(x) 0 .40 1 .25 2 .20 3 .05 4 .10 1.00 80/200

  15. There are four basic types of discrete probability distributions: 1. Uniform Distribution 2. Binomial Distribution 3. Poisson Distribution 4. Hyper-Geometric Distribution

  16. 5.1) Discrete Uniform Probability Distribution The probability distribution of a discrete probability distribution is given by the following formula. the values of the random variable are equally likely f(x) = 1/n where: n = the number of values the random variable may assume

  17. 5.2) Binomial Distribution • The probability distribution of a Binomial Distribution is given by the following function where: f(x) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial

  18. 5.3) Poisson Distribution • The probability of a Poisson Distribution is given by the following function where: f(x) = probability of x occurrences in an interval  = mean number of occurrences in an interval e = 2.71828

  19. for 0 <x<r 5.4) Hypergeometric Distribution • The probability of a Hyper-geometric Distribution is given by the following Function where: f(x) = probability of x successes in n trials n = number of trials N = number of elements in the population r = number of elements in the population labeled success

  20. 5.1) Discrete Uniform Probability Distribution The probability distribution of a discrete probability distribution is given by the following formula. the values of the random variable are equally likely f(x) = 1/n where: n = the number of values the random variable may assume

  21. Expected Value (Mean) and Variance of Discrete Probability Distributions

  22. E(x) =  = xf(x) Expected Value (Mean) for ……..Discrete Uniform Distribution The expected value, or mean, of a random variable is a measure of its central location.

  23. Var(x) =  2 = (x - )2f(x) Variance and Standard Deviation of Discrete Uniform Probability Distribution The variance summarizes the variability in the values of a random variable. The standard deviation, , is defined as the square root of the variance.

  24. Example-TV Sales in a Given Store • Given this data what is the Average Number of TVs sold in a day? • The likelihood of selling TV sets in any given day is considered equally likely (Uniform). The following table summarizes sales data on the past 200 days. xf(x) 0 .40 1 .25 2 .20 3 .05 4 .10 xF(x) 0 .40 1 .65 2 .85 3 .90 4 1.00 Number Units Soldof Days 0 80 1 50 2 40 3 10 4 20 200

  25. E(x) =  = xf(x) Expected Value (MEAN) and Variance • Given the data, we can find the Expected Value (Mean Number) of TVs sold in a day as follows xf(x)xf(x) 0 .40 .00 1 .25 .25 2 .20 .40 3 .05 .15 4 .10 .40 E(x) = 1.20 expected number of TVs sold in a day

  26. Var(x) =  2 = (x - )2f(x) Var(x) =  2 = (x - )2f(x)= 1.66 Find the Variance and Standard Deviation of the Number of TVs Sold in a given day. x (x - )2 f(x) (x - )2f(x) x -  -1.2 -0.2 0.8 1.8 2.8 1.44 0.04 0.64 3.24 7.84 0 1 2 3 4 .40 .25 .20 .05 .10 .576 .010 .128 .162 .784 TVs squared Standard deviation of daily sales = 1.2884 TVs

  27. 5.2) The Binomial Distribution

  28. 5.2) Binomial Distribution • A discrete probability distribution with binomial distribution has the following propertie • 1. The experiment consists of a sequence of n • identical trials. • 2. Only two outcomes, success and failure, are possible • on each trial. 3. The probability of a success, denoted by p, does not change from trial to trial. 4. The trials are independent.

  29. 5.2) The Binomial Distribution Typical Examples of a Binomial Experiment: • Lottery: Win or Lose • Election: A Candidate Wins or Loses • Gender of an Employee: is Male or Female • Flipping a coin: Heads or Tails

  30. 5.2) The Binomial Distribution Our interest is in the number of successes occurring in the n trials. let X denote the number of successes occurring in the n trials.

  31. 5.2) The Binomial Distribution • Binomial Probability Function where: n = the number of trials p = the probability of success on any one trial f(x) = the probability of x successes in n trials

  32. 5.2) The Binomial Distribution • Binomial Probability Function Probability of a particular sequence of outcomes with x successes Number of experimental outcomes providing exactly x successes in n trials

  33. Binomial Distribution • Example: Evans Electronics A local Electronics company is concerned about its low retention of employees. In recent years, management has seen an annual turnover of 10% in its hourly employees. Thus, for any hourly employee chosen at random, the company estimates that there is 0.1 probability that the person will leave the company in a year time.

  34. Binomial Distribution • Given the above information, if we randomly select 3 hourly employees, what is the probability that 1 of them will leave the company in one year ? Solution: Let: p = .10, n = 3, x = 1

  35. Binomial Distribution • Using Tables of Binomial Probabilities Page….592-600

  36. E(x) =  = np • Var(x) =  2 = np(1 -p) Mean and Variance of A Binomial Distribution • Expected Value • Variance • Standard Deviation

  37. E(x) =  = 3(.1) = .3 employees out of 3 Var(x) =  2 = 3(.1)(.9) = .27 Given that p=0.1, for the 3 randomly selected hourly employees, what is the expected number and variance of workers who might leave the company this year? • Expected Value • Variance • Standard Deviation

  38. 5.3) Poisson Distribution

  39. 6.4. Poisson Distribution • Poisson distribution refers to the probability distribution of a trial that involves cases of rare events that occur over a fixed timeinterval or within aspecified region

  40. 6.4. Poisson Distribution Examples…. • The number of errors a typist makes per page • The number of cars entering a service station per hour • The number of telephone calls received by a switchboard per hour. • The number of bank failures during a given economic recession. • The number of housing foreclosures in a given city during a given year. • The number of car accidents in one day on I-35 stretch from the Twin Cities to Duluth

  41. Poisson Distribution We use a Poisson distribution to estimate the number of occurrences of these discrete events that often occur over a specified interval of time or space That is, a Poisson distributed random variable is discrete; Often times it assumes an infinite sequence of values (x = 0, 1, 2, . . . ).

  42. Properties of a Poisson Experiment • The number of successes (events) that occur in a certain time interval is independent of the number of successes that occur in another time interval. • The probability 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. • The probability that two or more successes will occur in an interval approaches zero as the interval becomes smaller.

  43. Poisson Distribution • Poisson Probability Function where: f(x) = probability of x occurrences in an interval  = mean number of occurrences in an interval e = 2.71828

  44. Poisson Distribution--Example On average 6 patients arrive per hour at Mercy Hospital emergency room on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? MERCY

  45. MERCY Poisson Distribution-Example • Using the Poisson Probability Function • = 6/hour = 3/half-hour, P(x = 4)?

  46. MERCY Poisson Distribution • Using Poisson Probability Tables Pages: 602-607

  47. m = s 2 Mean and Variance of Poisson Distribution • Another special property of the Poisson distribution • is that the mean and variance are equal.

  48. MERCY Poisson Distribution • Variance for Number of Arrivals During 30-Minute Periods m = s2 = 3

  49. 5.4) Hyper-Geometric Distribution

  50. Hyper-geometric Distribution The hyper-geometric distribution is closely related to a binomial distribution. However, for the hyper-geometric distribution: the trials are not independent, and the probability of success changes from trial to trial.

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