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Discrete probability Business Statistics (BUSA 3101) Dr. Lari H. Arjomand lariarjomand@clayton

Discrete probability Business Statistics (BUSA 3101) Dr. Lari H. Arjomand lariarjomand@clayton.edu. Discrete ProbabilityDistributions. .40. .30. .20. .10. 0 1 2 3 4. Random Variables. Discrete Probability Distributions. Expected Value and Variance.

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Discrete probability Business Statistics (BUSA 3101) Dr. Lari H. Arjomand lariarjomand@clayton

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  1. Discrete probability Business Statistics (BUSA 3101)Dr. Lari H. Arjomandlariarjomand@clayton.edu

  2. Discrete ProbabilityDistributions .40 .30 .20 .10 0 1 2 3 4 • Random Variables • Discrete Probability Distributions • Expected Value and Variance • Binomial Distribution • Poisson Distribution (Optional Reading) • Hypergeometric Distribution (Optional Reading)

  3. Random Variables • A random variable is a numerical description of the outcome of an experiment. • A discrete random variable may assume either a finite number of values or an infinite sequence of values. • A continuous random variable may assume any numerical value in an interval or Cllection of intervals.

  4. Example: JSL Appliances Discrete random variable with a finite number of values Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4)

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

  6. Random Variables Examples Type Question Random Variable x Family size x = Number of dependents reported on tax return Discrete Continuous x = Distance in miles from home to the store site Distance from home to store Own dog or cat Discrete x = 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)

  7. Random VariablesDefinition & Example Definition: Arandom variable is a quantity resulting from a random experiment that, by chance, can assume different values. Example: Consider a random experiment in which a coin is tossed three times. Let X be the number of heads. Let H represent the outcome of a head and T the outcome of a tail.

  8. The sample spacefor such an experiment will be: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH. Thus the possible values of X (number of heads) are X = 0, 1, 2, 3. This association is shown in the next slide. Note:In this experiment, there are 8 possible outcomes in the sample space. Since they are all equally likely to occur, each outcome has a probability of 1/8 of occurring. Example (Continued)

  9. Example (Continued) TTT TTH THT THH HTT HTH HHT HHH 0 1 1 2 1 2 2 3 Sample Space X

  10. The outcome of zero heads occurred only once. The outcome of one head occurred three times. The outcome of two heads occurred three times. The outcome of three heads occurred only once. From the definition of a random variable, X as defined in this experiment, is a random variable. X values are determined by the outcomes of the experiment. Example (Continued)

  11. Probability Distribution: Definition Definition: A probability distribution is a listing of all the outcomes of an experiment and their associated probabilities. The probability distribution for the random variable X (number of heads) in tossing a coin three times is shown next.

  12. Probability Distribution for Three Tosses of a Coin

  13. Data Types

  14. Discrete Random Variable Examples

  15. Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. We can describe a discrete probability distribution with a table, graph, or equation.

  16. Discrete Probability Distributions • The probability distribution is defined by a • probability function, denoted by f(x), which provides • the probability for each value of the random variable. • The required conditions for a discrete probability function are: f(x) > 0 P(X) ≥ 0 ΣP(X) = 1 f(x) = 1

  17. Discrete Probability Distributions Example • Using past data on TV 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

  18. Graphical Representation of Probability Distribution .50 .40 .30 .20 .10 Discrete Probability Distributions Probability 0 1 2 3 4 Values of Random Variable x (TV sales)

  19. As we said, the probability distribution of a discrete random variable is a table, graph, or formula that gives the probability associated with each possible value that the variable can assume. Example : Number of Radios Sold at Sound City in a Week x, Radios p(x), Probability 0 p(0) = 0.03 1 p(1) = 0.20 2 p(2) = 0.50 3 p(3) = 0.20 4 p(4) = 0.05 5 p(5) = 0.02

  20. Expected Value of a Discrete Random Variable • The meanorexpected value of a discrete random • variable is: Example: Expected Number of Radios Sold in a Week x, Radios p(x), Probability x p(x) 0 p(0) = 0.03 0(0.03) = 0.00 1 p(1) = 0.20 1(0.20) = 0.20 2 p(2) = 0.50 2(0.50) = 1.00 3 p(3) = 0.20 3(0.20) = 0.60 4 p(4) = 0.05 4(0.05) = 0.20 5 p(5) = 0.025(0.02) = 0.10 1.00 2.10

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