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

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

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

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

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

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

Mean of Discrete 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 .  x P ( x )  Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Variance of Discrete 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.   ( x –  ) 2 P ( x )  Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Discrete Uniform Random 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

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

Mean and Variance of a Binomial Random Variable   np (1– p )   Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

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

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

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

Mean and Variance for the Hypergeometric Distribution  2  x 2 P ( x )–  2   n k N       1– k N               N – n N –1       Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

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

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

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

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

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

Mean and Variance of the Poisson Distribution   xPxP ( x )  Mean of X  ( x –  ) 2  P ( x )  Variance of X   Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing