Discrete Probability Distribution Probability Distribution of a

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

• 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*P(X) 0 40 0. 08 0 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 8 0. 016 0. 16 48. 219136 0. 771506176 11 2 0. 004 0. 044 63. 107136 0. 252428544 Total 500 Mean = 3. 056 Variance = Std Dev = 6. 068864 2. 463506444 X is the number of cars sold per day; P(X) is the probability 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. 284 + 0. 132) = 0. 696 • P(X 4) = P(X < 4) + P(X = 4) = 0. 696 + 0. 072 = 0. 768 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 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 • – A firm bidding for contracts will either get a contract or not 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).

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 3. n! =n(n - 1)(n - 2). . . (2)(1) X! = X(X - 1)(X - 2). . . (2)(1) 0! = 1 (by definition) 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, Not OL, and OL. 2. How many sequences (order) of 3 over the limit are possible? 3. 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 1. 2. 3. 4. 5. 6. 7. 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 = 8. 9. 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). 1. http: //wps. prenhall. com/wps/media/objects/9431/9657451/Ch_05/levine-smume 6_topic_BINO. pdf

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