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. Random Variables Probability Distribution of a Discrete Random Variable Probability Distribution of a Continuous Random Variable Ch.6 Ch.5

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.

Expected Value (or mean) of a discrete distribution (Weighted Average) Variance of a discrete random variable Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X=Mean Xi = the ith outcome of the variable X P(Xi) = Probability of the Xi occurrence

Number of Cars Sold, X Frequency P(X) X*P(X) [X-E(X)]^2 [X-E(X)]^2*P(X) 40 0.08 9.339136 0.74713088 1 100 0.2 4.227136 0.8454272 2 142 0.284 0.568 1.115136 0.316698624 3 66 0.132 0.396 0.003136 0.000413952 4 36 0.072 0.288 0.891136 0.064161792 5 30 0.06 0.3 3.779136 0.22674816 6 26 0.052 0.312 8.667136 0.450691072 7 20 0.04 0.28 15.555136 0.62220544 8 16 0.032 0.256 24.443136 0.782180352 9 14 0.028 0.252 35.331136 0.989271808 10 0.016 0.16 48.219136 0.771506176 11 0.004 0.044 63.107136 0.252428544 Total 500 Mean = 3.056 Variance = 6.068864 Std Dev = 2.463506444 X is the number of cars sold per day; P(X) is the probability that that many are sold per day.

NOTE: The usefulness of the Table P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = (0.08 + 0.2 + 0.284 + 0.132) = 0.696 P(X 4) = P(X < 4) + P(X = 4) = 0.696 + 0.072 = 0.768 P(X 4) = 1 – P(X < 4) = 1 – 0.696 = 0.304 P(X = 4) = 0.072 P(X > 4) = 1 – P(X 4) = 1 – 0.768 = 0.232 Go to handout example

Binomial Probability Distribution (a special discrete distribution) Characteristics A fixed number of identical observations, n. Each observation is drawn from: Infinite population without replacement or Finite population with replacement Two mutually exclusive (?) and collectively exhaustive (?) categories Generally 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 other The outcome of one observation does not affect the outcome of the other

Binomial Distribution has many application in business Examples: A firm bidding for contracts will either get a contract or not A manufacturing plant labels items as either defective or acceptable A 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 it An account is either delinquent or not Example: 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).

2. Number of possible sequences (or orders) Let, p = probability of “success” in one trial or observation n = 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 trial Then, 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)

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.

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 symmetrical If p< 0.5, the distribution is right skewed If p>0.5, the distribution is left skewed The closer p to 0.5 and the larger the sample size, n, less skewed the distribution The 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). http://wps.prenhall.com/wps/media/objects/9431/9657451/Ch_05/levine-smume6_topic_BINO.pdf