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

Discrete ProbabilityDistributions n Random Variables n Discrete Probability Distributions n Expected Value and Variance n Binomial Distribution n Poisson Distribution (Optional Reading) n Hypergeometric Distribution ( Optional Reading )

Random Variables 1.A random variable is a numerical description of the outcome of an experiment. 2.A discrete random variable may assume either a finite number of values or an infinite sequence of values. 3.A continuous random variable may assume any numerical value in an interval or Cllection of intervals.

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

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

Random Variables ExamplesQuestion Random Variable x Type Familysize x = Number of dependents reported on tax return reported on tax returnDiscrete Distance from home to store x = Distance in miles from home to the store site home to the store site Continuous Own dog or cat x = 1 if own no pet; = 2 if own dog(s) only; = 2 if own dog(s) only; = 3 if own cat(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) = 4 if own dog(s) and cat(s) Discrete

Random Variables Definition & Example Definition: A Definition: A random 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. 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.

The for such an experiment will be: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH. The sample space for 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. Thus the possible values of X (number of heads) are X = 0, 1, 2, 3. This association is shown in the next slide. 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. 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)

TTTTTHTHTTHHHTTHTHHHTHHH TTTTTHTHTTHHHTTHTHHHTHHH Sample Space X Example (Continued)

The outcome of zero heads occurred only once. The outcome of zero heads occurred only once. The outcome of one head occurred three times. The outcome of one head occurred three times. The outcome of two heads occurred three times. The outcome of two heads occurred three times. The outcome of three heads occurred only once. 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. 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. X values are determined by the outcomes of the experiment. Example (Continued)

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

Probability Distribution for Three Tosses of a Coin

Data Types

Discrete Random Variable Examples

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

Discrete Probability Distributions f ( x ) > 0  f ( x ) = 1 P(X) ≥ 0 ΣP(X) = 1  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:

n a tabular representation of the probability distribution for TV sales was developed. distribution for TV sales was developed. n Using past data on TV sales, … Number Number Units Sold of Days Units Sold of Days x f ( x ) x f ( x ) /200 Discrete Probability Distributions Example

Values of Random Variable x (TV sales) ProbabilityProbability Discrete Probability Distributions  Graphical Representation of Probability Distribution

 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, Radiosp(x), Probability 0p(0) = p(1) = p(2) = p(3) = p(4) = p(5) = 0.02

Expected Value of a Discrete Random Variable  The mean or expected value of a discrete random variable is: Example: Expected Number of Radios Sold in a Week x, Radiosp(x), Probabilityx p(x) 0p(0) = 0.030(0.03) = p(1) = 0.201(0.20) = p(2) = 0.502(0.50) = p(3) = 0.203(0.20) = p(4) = 0.054(0.05) = p(5) = 0.025(0.02) =