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## Discrete Random Variables

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**Discrete Random Variables**• Discrete Probability Distributions • Binomial Distribution • Poisson Distribution • Hypergeometric Distribution Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Discrete Random Variables – Sample Space includes all mutually exclusive outcomes Probabilities – from subjective, frequency or subjective methods Two conditions apply: Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Probability distributions can be estimated from relative frequencies. Consider the discrete (countable) number of televisions per household (millions) from US survey data: Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Probability distributions can be estimated from relative frequencies. Consider the discrete (countable) number of televisions per household from US survey data: 1,218 ÷ 101,501 = 0.012 Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Probability distributions can be estimated from relative frequencies. Consider the discrete (countable) number of televisions per household from US survey data: EX: P(X=4) = P(4) = 0.076 = 7.6% Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**What is the probability there is at least one television but no more than three in any given household? Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**What is the probability there is at least one television but no more than three in any given household? Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**What is the probability there is at least one television but no more than three in any given household? “at least one television but no more than three” P(1 ≤ X ≤ 3) = P(1) + P(2) + P(3) = .319 + .374 + .191 = .884 Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Assume a mutual fund salesman knows that there is 20% chance of closing a sale on each call he makes. Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**What is the probability distribution of the number of sales if he plans to call three customers? Let S denote success, making a sale: P(S) = .20, then Sʹ, not making a sale: P(Sʹ) = .80 Econ10/Mgt 10 Stuffler**P(S)=.2**P(S’)=.8 Discrete Probability Distributions • Developing a Probability Distribution Tree Sales Call 1 Econ10/Mgt 10 Stuffler**P(S)=.2**P(S)=.2 P(S’)=.8 P(S)=.2 P(S’)=.8 P(S’)=.8 Discrete Probability Distributions • Developing a Probability Distribution Tree Sales Call 1 Sales Call 2 Econ10/Mgt 10 Stuffler**P(S)=.2**P(S)=.2 P(S’)=.8 P(S)=.2 P(S)=.2 P(S’)=.8 P(S’)=.8 P(S)=.2 P(S)=.2 P(S’)=.8 P(S’)=.8 P(S)=.2 P(S’)=.8 P(S’)=.8 Discrete Probability Distributions • Developing a Probability Distribution Tree Sales Call 1 Sales Call 2 Sales Call 3 S S S S S S’ S S’S S S’S’ S’S S S’S S’ S’S’S S’S’S’ Econ10/Mgt 10 Stuffler**P(S)=.2**P(S)=.2 P(S’)=.8 P(S)=.2 P(S)=.2 P(S’)=.8 P(S’)=.8 P(S)=.2 P(S)=.2 P(S’)=.8 P(S’)=.8 P(S)=.2 P(S’)=.8 P(S’)=.8 Discrete Probability Distributions • Developing a Probability Distribution Tree Sales Call 1 Sales Call 2 Sales Call 3 S S S S S S’ S S’S S S’S’ S’S S S’S S’ S’S’S S’S’S’ Econ10/Mgt 10 Stuffler**P(S)=.2**P(S)=.2 P(S’)=.8 P(S)=.2 P(S)=.2 ‘)=.8 P(S’)=.8 P(S)=.2 P(S)=.2 P(S’)=.8 P(S’)=.8 P(S)=.2 P(S’)=.8 P(S’)=.8 Discrete Probability Distributions • Developing a Probability Distribution Sales Call 1 Sales Call 2 Sales Call 3 S S S S S S’ S S’S S S’S’ S’S S S’S S’ S’S’S S’S’S’ • X P(x) • .23 = .008 • 3(.032)=.096 • 3(.128)=.384 • 0 .83 = .512 Econ10/Mgt 10 Stuffler**P(S)=.2**P(S)=.2 P(SC)=.8 P(S)=.2 P(S)=.2 P(SC)=.8 P(SC)=.8 P(S)=.2 P(S)=.2 P(SC)=.8 P(SC)=.8 P(S)=.2 P(SC)=.8 P(SC)=.8 Discrete Probability Distributions • Explain how to derive .032 and .128 Sales Call 1 Sales Call 2 Sales Call 3 S S S S S SC S SC S S SC SC SC S S SC S SC SC SC S SC SC SC • X P(x) • .23 = .008 • 3(.032)=.096 • 3(.128)=.384 • 0 .83 = .512 Econ10/Mgt 10 Stuffler**P(S)=.2**P(S)=.2 P(SC)=.8 P(S)=.2 P(S)=.2 P(SC)=.8 P(SC)=.8 P(S)=.2 P(S)=.2 P(SC)=.8 P(SC)=.8 P(S)=.2 P(SC)=.8 P(SC)=.8 Discrete Probability Distributions • The P(X=2) is: Sales Call 1 Sales Call 2 Sales Call 3 (.2)(.2)(.8)= .032 S S S S S SC S SC S S SC SC SC S S SC S SC SC SC S SC SC SC • X P(x) • .23 = .008 • 3(.032)=.096 • 3(.128)=.384 • 0 .83 = .512 Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**• A discrete probability distribution represents a population Example: Population of number of TVs per household Example: Population of sales call outcomes • Since we have populations, we can describe • them by computing various parameters: • Population Mean and Population Variance Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**• Population Mean (Expected Value) • - Weighted average of all values with the weights • being the probabilities • - Expected value of X, E(X) Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Population variance - weighted average of the squared deviations from the mean. “Short cut” formula for the variance Standard deviation formula Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Find the mean, variance, and standard deviation for the population of the number of color televisions per household. = 0(.012) + 1(.319) + 2(.374) + 3(.191) + 4(.076) + 5(.028) = 2.084 Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Find the mean, variance, and standard deviation for the population of the number of color televisions per household. = (0 – 2.084)2(.012) + (1 – 2.084)2(.319)+…+(5 – 2.084)2(.028) = 1.107 Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Find the mean, variance, and standard deviation for the population of the number of color televisions per household. = 1.052 Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Special application of Expected Value Suppose the probability that an insurance agent makes a sale is .20 and after costs earns a commission of $525. If he/she does not make a sale, they must pay $75 in costs. What is their expected value from a sales call? Does the benefit exceed the cost or is the opposite true? Econ10/Mgt 10 Stuffler**Discrete Probability Distributions**Special application of Expected Value Let X be the discrete random variable of making a sale call x P(x) x•P(x) Sale $ 525 .20 $ 105 No Sale - $ 75 .80 - $ 60 E(x) = μ = ∑ xP(x) = $ 45 Econ10/Mgt 10 Stuffler**Binomial Distribution**• Binomial distribution is the probability distribution that results from doing a “binomial experiment” which have the properties: • Fixed number of identical trials, represented as n. • Each trial has two possible outcomes: a “success” or “failure” • For all trials, the probability of success, P(success)=p, and the probability of failure, P(failure)=1–p=q, are constant. • The trials are independent Econ10/Mgt 10 Stuffler**Several Binomial Distributions**Econ10/Mgt 10 Stuffler**Binomial Distribution**• Success and failure: labels for binomial experiment outcomes, no value judgment is implied. • EX: Coin flip results in either heads or tails. • If we define “heads” as success, then • “tails” is considered a failure. • Other binomial examples: • An election candidate wins or loses • An employee is male or female • A worker is employed or unemployed Econ10/Mgt 10 Stuffler**Binomial Distribution**Binomial Formula: where x = number of successes in n trials, n – x = number of failures in n trials, px = the probability of success raised to the number of successes, and qn-x = probability of failure raised to the number of failures Econ10/Mgt 10 Stuffler**Binomial Distribution**• The random variable of a binomial experiment is defined as the number of successes in the n trials, and is called the binomial random variable. • EX: Flip a fair coin 10 times • 1) Fixed number of trials n=10 • 2) Each trial has two possible outcomes {heads (success), • tails (failure)} • 3) P(success)= 0.50; P(failure)=1–0.50 = 0.50 • 4) The trials are independent (i.e. the outcome of heads on the • first flip will have no impact on subsequent coin flips). • Flipping a coin ten times is a binomial • experiment since all conditions are met. Econ10/Mgt 10 Stuffler**Binomial Distribution**Another sales call example: Assume a mutual fund salesman knows that there is • 20% chance of closing a sale on each call he makes. We want to determine the probability of making two sales in three calls: P(sale) = .2 P(no sale) = .8 P(X=2) = 3! = .096 2!(3-2)! .22.8.8 Econ10/Mgt 10 Stuffler**Binomial Distribution**Another sales call example: Assume a mutual fund salesman knows that there is 20% chance of closing a sale on each call he makes We want probability of making two sales in three calls P(sale) = .2 P(no sale) = .8 GOOD NEWS! Megastat→Probability→Discrete Distributions→Binomial Econ10/Mgt 10 Stuffler**Binomial Distribution**Econ10/Mgt 10 Stuffler**Pat Statsly**• Pat Statsly is a student (not a good student) taking a • statistics course. Pat’s exam strategy is to rely on luck • for the first test. The test consists of 10 multiple-choice • questions. Each question has five possible answers, only • one of which is correct. Pat plans to guess the answer • to each question. • What is the probability that Pat gets no answers correct? • What is the probability that Pat gets two answers correct? Econ10/Mgt 10 Stuffler**Pat Statsly**n=10 P(correct) = p = 1/5 = .20 P(wrong) = q = .80 Is this a binomial experiment? Check the conditions Econ10/Mgt 10 Stuffler**Pat Statsly**n=10 P(correct) = p = 1/5 = .20 P(wrong) = q = .80 Is this a binomial experiment?Check the conditions: There is a fixed finite number of trials (n=10). An answer can be either correct or incorrect. The probability of a correct answer (P(success)=.20) does not change from question to question. Each answer is independent of the others. Econ10/Mgt 10 Stuffler**Pat Statsly**• n=10, and P(success) = .20 • What is the probability that Pat gets no answers correct? • EX: P(x=0) What’s the interpretation of this result? Econ10/Mgt 10 Stuffler**Pat Statsly**• n=10, and P(success) = .20 • What is the probability that Pat gets no answers correct? • EX: P(x=0) Pat has about an 11% chance of getting no answers correct using the guessing strategy. Econ10/Mgt 10 Stuffler**Pat Statsly**Econ10/Mgt 10 Stuffler**Pat Statsly**• n=10, and P(success) = .20 • What is the probability that Pat gets two answers correct? • EX: P(x=2) Econ10/Mgt 10 Stuffler**Pat Statsly**• We have been using the binomial probability distribution to find probabilities for individual values of x. • To answer the question: • “Find the probability that Pat fails the quiz” • requires a cumulative probability, that is, P(X ≤ x) • If a grade on the quiz is less than 50% (i.e. 5 questions • out of 10), that’s considered a failed quiz. Econ10/Mgt 10 Stuffler**Pat Statsly**• We have been using the binomial probability distribution to find probabilities for individual values of x. • To answer the question: • “Find the probability that Pat fails the test” • requires a cumulative probability, that is, P(X ≤ x) • If a grade on the test is less than 50% (i.e., 5 questions • out of 10), that’s considered a failed test. • We want to know what is: P(X ≤ 4) to answer Econ10/Mgt 10 Stuffler**Pat Statsly**P(X ≤ 4) = P(0) + P(1) + P(2) + P(3) + P(4) = .9672 What is the interpretation of this result? Econ10/Mgt 10 Stuffler**Pat Statsly**Its about 97% probable that Pat will fail the test using the luck strategy and guessing at answers. Econ10/Mgt 10 Stuffler**Binomial Table**• Calculating binomial probabilities by hand is tedious and • error prone. There is an easier way. Refer to Table E-6 in • the Appendices. For the Pat Statsly example,n=10, so go to • the n=10 table: • Look in the Column, p=.20 and substitute into: • P(X ≤ 4) = P(0) + P(1) + P(2) + P(3) + P(4) • P(X ≤ 4) = .1074 + .2684 + .3020 + .2013 + .0881 = .9672 • The probability of Pat failing the test is 96.72% Econ10/Mgt 10 Stuffler**Poisson Distribution**• Named for Simeon Poisson, Poisson distribution - • Discrete probability distribution • There there are no trials • Number of independent events (successes) occurring in a fixed time period or region of space that occur with a known average rate such as, arrivals, departures, or accidents, number of baskets in a quarter, etc. Econ10/Mgt 10 Stuffler**Poisson Distribution**• For example: • The number of cars arriving at a service station in 1 hour. (The interval of time is 1 hour.) • The number of flaws in a bolt of cloth. (The specific region is a bolt of cloth.) • The number of accidents in 1 day on a particular stretch of highway. (The interval is defined by both time, 1 day, and space, the particular stretch of highway.) Econ10/Mgt 10 Stuffler**Poisson Distribution**• Poisson random variable - number of successes that occur in a period of time or an interval of space • EX: On average, 96 trucks arrive at a border crossing • every hour. • EX: Number of typographical errors in a new textbook • edition averages 1.5 per 100 pages. Econ10/Mgt 10 Stuffler**Poisson Distribution…**• The Poisson random variable is the number of successes that occur in a period of time or an interval of space in a Poisson experiment. • EX: On average, 96 trucks arrive at a border crossing • every hour. • E.g. The number of typographic errors in a new textbook edition averages 1.5 per 100 pages. successes time period successes (?!) interval Econ10/Mgt 10 Stuffler**Poisson Distribution**Poisson experiment has four defining characteristics or properties: The number of successes that occur in any interval is independent of the number of successes that occur in any other interval The probability of a success in an interval is the same for all equal-size intervals The probability of a success is proportional to the size of the interval Only one value is required to determine the probability of a designated number of events occurring during an interval Econ10/Mgt 10 Stuffler