Discrete Probability Distributions

1. Discrete Probability Distributions Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

2. Random Variable A Random Variable is a function that assigns a numerical value to each outcome of an experiment. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

3. Discrete Random Variable A discrete random variable is a random variable whose values are counting numbers or discrete data. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

4. Continuous Random Variable A continuous random variable is a random variable for which any value is possible over some continuous range of values. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

5. Example 5.2 Consider a discrete random variable X having possible values of 1, 2, or 3. The corresponding probability for each value is:  1 with probability  X =  2 with probability   3 with probability  Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

6. Example 5.2 Consider the function: P(X = x) = P(x) = x/6 or X P(X) 1  2  3  1 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

7. Mean ofDiscrete Random Variables The mean of a discrete random variable represents the average value of the random variable if you were to observe this variable over an indefinite period of time. The mean of a discrete random variable is written as . m = å x P ( x ) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

8. Variance ofDiscrete Random Variable The variance of a discrete random variable, X, is a parameter describing the variation of the corresponding population. The symbol used is 2 . 2 å ( x – m ) P ( x ) s2 = Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

9. Discrete UniformRandom Variable A discrete uniform random variable has the property that it is discrete and that its values all have the same probability of occurring. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

10. Binomial Random Variable • The experiment consists of n repetitions, called trials. • Each trial has two mutually exclusive possible outcomes, referred to as success and failure. • The n trials are independent. • The probability for a success for each trial is denoted p; and remains the same for each trial • The random variable X is the number of successes out of n trials. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

11. Mean and Variance of a Binomial Random Variable m = np s2 = np ( 1 – p ) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

12. Using the Binomial Table to determine Binomial Probabilities The binomial PMFs have been tabulated in Table A.1 for various values of n and p. If n = 4 and p = 0.3 and you wish to find the P(2) locate n = 4 and x=2. Go across to p = 0.3 and you will find the corresponding probability (after inserting the decimal in front of the number). This probability is 0.265. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

13. Hypergeometric Distribution • The hypergeometric distribution bears a strong resemblance to the binomial random variable. • The experiment consists of n trials • Has two possible outcomes • Primary distinction between the Hypergeometric and the Binomial is that the trials in the Hypergeometric are not independent. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

14. Conditions for a Hypergeometric Distribution • Population size = N k members are S (successes) and N-k are F(failures) • Sample size = n trials obtained without replacement • X = the number of successes Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

15. Mean and Variance for theHypergeometric Distribution 2 2 2 x P ( x ) – m å s = é k k ù N – n æ ö æ ö æ ö = n 1 – ê ç ÷ ç ÷ ú ç ÷ ê è N ø è N ø ú è N – 1 ø ë û Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

16. Hypergeometric Probabilities Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

17. The Poisson Distribution The poisson distribution is useful for counting the number of times a particular event occurs over a specified period of time or over a specified area. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

18. Conditions for the Poisson Distribution • The Number of Occurrences in One Measurement Unit are independent of the Number of Occurrences in any other other Non-Overlapping Measurement Unit. • The Expected Number of Occurrences in any given Measurement Unit are proportional to the size of the Measurement Unit. • Events can not occur at exactly the same point in the Measurement Unit. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

19. Examples of Poisson Measurement Units • Time: • Arrivals of customers at a service facility • Requests for replacement parts • Linear: • Defects in linear feet of a spool of wire • Defects in square yards of carpet Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

20. Poisson Probability Mass Function x m e m – P ( x ) = x ! for x = 0 , 1, 2, 3, . . . Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

21. Mean and Variance of the Poisson Distribution m = å xP ( x ) = Mean of X 2 m = ( x – m ) × P ( x ) Variance of X = å Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing