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PROBABILITY DISTRIBUTIONS Chapter 5 Outline Section 5-1: Expected Value Section 5-2: Binomial Distribution Section 5-3: Poisson Distribution (OMIT) Section 5-4: Hypergeometric Distribution (OMIT) Introduction Overview This chapter will deal with the construction of probability distributions by combining methods of descriptive statistics from Chapters 2 and 3 and those of probability presented in Chapter 4. A probability distribution, in general, will describe what will probably happen instead of what actually did happen Combining Descriptive Methods and Probabilities In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect. Why do we need probability distributions? Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results Saleswoman can compute probability that she will make 0, 1, 2, or 3 or more sales in a single day. Then, she would be able to compute the average number of sales she makes per week, and if she is working on commission, she will be able to approximate her weekly income over a period of time. An investor wants to compare the risks of two different stock options for his portfolio Probability Distributions Remember From Chapter 1, a variable is a characteristic or attribute that can assume different values Represented by various letters of the alphabet From Chapter 1, a random variable is a variable whose values are determined by chance Typically assume values of 0,1,2…n Remember Discrete Variables (Data)— Chapter 5 Can be assigned values such as 0, 1, 2, 3 “Countable” Examples: Number of children Number of credit cards Number of calls received by switchboard Number of students Continuous Variables (Data)--Chapter 6 Can assume an infinite number of values between any two specific values Obtained by measuring Often include fractions and decimals Examples: Temperature Height Weight Time Examples: State whether the variable is discrete or continuous 1) 2) 3) 4) 5) 6) 7) The height of a randomly selected giraffe living in Kenya The number of bald eagles located in New York State The exact time it takes to evaluate 27 + 72 The number of textbook authors now sitting at a computer The exact life span of a kitten The number of statistics students now reading a book The weight of a feather Discrete Probability Distribution Consists of the values a random variable can assume and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation Can be shown by using a graph (probability histogram), table, or formula Two requirements: The probability of each event in the sample space must be between or equal to 0 and 1. That is, 0 < P(x) < 1 The sum of the probabilities of all the events in the sample space must equal 1; that is, SP(x) = 1 Example: Determine whether the distribution represents a probability distribution. If it does not, state why. 8) 9) x 3 6 8 12 x 1 2 3 4 5 P(x) 0.3 0.5 0.7 -0.8 P(x) 0.3 0.1 0.1 0.2 0.3 Example: Determine whether the distribution represents a probability distribution. If it does not, state why. 10) A researcher reports that when groups of four children are randomly selected from a population of couples meeting certain criteria, the probability distribution for the number of girls is given in the accompanying table x P(x) 0 0.502 1 0.365 2 0.098 3 0.011 4 0.001 Section 5.1 Expected Value Objective: Calculate the expected value of a probability distribution Once we know that a probability distribution exist, we can describe it using various descriptive statistics Visually using a graph, table, or formula Algebraically, we can find the mean, variance, and standard deviation Mean of a general discrete probability distribution m Sxp( x) x1 p( x1 ) x2 p( x2 ) ...xn p( xn ) m= population mean since ALL possible values are considered Mean is also known as “Expected Value” Mean should be rounded to one more decimal place than the outcome x. Always simplify fractions Variance & standard deviation 2 ( x 2 p( x)) m 2 Variance ( ( x 2 p( x )) m 2 Standard Deviation Example –Use table on example 9 to find mean and standard deviation x 1 2 3 4 5 P(x) 0.3 0.1 0.1 0.2 0.3 Assignment Worksheet ---Section 5.1 Answers to Examples 1) 2) 3) 4) 5) 6) 7) 8) Continuous Discrete Continuous Discrete Continuous Discrete Continuous This is not a probability distribution because one of the probabilities is negative (is not between 0 and 1)