Presentation on theme: "Discrete Probability Distribution"— Presentation transcript:
1 Discrete Probability Distribution Probability Distribution of a Random Variable: Is a table, graph, or mathematical expression that specifies all possible values (outcomes) of a random variable along with their respective probabilities.RandomVariablesProbability Distribution of a DiscreteRandom VariableProbability Distribution of a ContinuousRandom VariableCh.6Ch.5
2 A discrete probability distribution applied to countable values (That is to a random variables resulting from counting, not measuring)Example: Using the records for past 500 working days, a manager of auto dealership summarized the number of cars sold per day and the frequency of each number sold.Questions:What is the average number of cars sold per day?What is the dispersion of the number of cars sold per day?What is the probability of selling less than 4 cars per day?What is the probability of selling exactly 4 cares per day?What is the probability of selling more than 4 cars per day?To answer these questions, we need the mean and the standard deviation of the distribution.
3 Expected Value (or mean) of a discrete distribution (Weighted Average) Variance of a discrete random variableStandard Deviation of a discrete random variablewhere:E(X) = Expected value of the discrete random variable X=MeanXi = the ith outcome of the variable XP(Xi) = Probability of the Xi occurrence
4 Number of Cars Sold, XFrequencyP(X)X*P(X)[X-E(X)]^2[X-E(X)]^2*P(X)400.0811000.221420.2840.5683660.1320.3964360.0720.2885300.060.36260.0520.3127200.040.288160.0320.2569140.0280.252100.0160.16110.0040.044Total 500Mean =3.056Variance =Std Dev =X is the number of cars sold per day; P(X) is the probability that that many are sold per day.
6 Binomial Probability Distribution (a special discrete distribution) CharacteristicsA fixed number of identical observations, n. Each observation is drawn from:Infinite population without replacement orFinite population with replacementTwo mutually exclusive (?) and collectively exhaustive (?) categoriesGenerally called “success” and “failure”Probability of success is p, probability of failure is (1 – p)Constant probability for each outcome from one observation to observation over all observations.Observations are independent from each otherThe outcome of one observation does not affect the outcome of the other
7 Binomial Distribution has many application in business Examples: A firm bidding for contracts will either get a contract or notA manufacturing plant labels items as either defective or acceptableA marketing research firm receives survey responses of “yes I will buy” or “no I will not”New job applicants either accept the offer or reject itAn account is either delinquent or notExample: Suppose 4 credit card accounts are examined for over the limit charges. Overall probability of over the limit charges is known to be 10 percent (one out of every 10 accounts).
8 2. Number of possible sequences (or orders) Let,p = probability of “success” in one trial or observationn = sample size (number of trials or observations)X = number of ‘successes’ in sample, (X = 0, 1, 2, ..., n)P(X) = probability of X successes in n trials, with probability of success p on each trialThen,1. P(X success in a particular sequence (or order) =2. Number of possible sequences (or orders)Where n! =n(n - 1)(n - 2) (2)(1)X! = X(X - 1)(X - 2) (2)(1)0! = 1 (by definition)3. P(X success regardless of the sequence or order)
9 Back to our example:What is the probability of 3 account being over the limit with the following order? OL,OL,Not OL, and OL.2. How many sequences (order) of 3 over the limit are possible?What is the probability of 3 accounts being over the limit in all possible orders? (all Possible sequences)Solutions: Calculate 1, 2, and 3.
10 Characteristics of a Binomial Distribution For each pair of n and p a particular probability distribution can be generated.The shape of the distribution depends on the values of p and n.If p=0.5, the distribution is perfectly symmetricalIf p< 0.5, the distribution is right skewedIf p>0.5, the distribution is left skewedThe closer p to 0.5 and the larger the sample size, n, less skewed the distributionThe mean of the distribution =The standard deviation =Or you can download the binomial table on the text- book’s companion Web site. (You need to learn how to use this table for the test).