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Sections 4.1 and 4.2 Overview Random Variables

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PROBABILITY DISTRIBUTIONS This chapter will deal with the construction of probability distributions by combining the methods of Chapter 2 with the those of Chapter 3. Probability Distributions will describe what will probably happen instead of what actually did happen.

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COMBINING DESCRIPTIVE METHODS AND PROBABILITIES In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect.

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RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure. A probability distribution is a graph, table, or formula that gives the probability for each value of a random variable.

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EXAMPLES 1.Suppose you toss a coin three times. Let x be the total number of heads. Make a table for the probability distribution of x. 2.Suppose you throw a pair of dice. Let x be the sum of the numbers on the dice. Make a table for the probability distribution of x.

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SAMPLE SPACE FOR ROLLING A PAIR OF DICE 1-11-21-31-41-51-6 2-12-22-32-42-52-6 3-13-23-33-43-53-6 4-14-24-34-44-54-6 5-15-25-35-45-55-6 6-16-26-36-46-56-6

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DISCRETE AND CONTINUOUS RANDOM VARIABLES A discrete random variable has either a finite number of values or a countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they can be associated with a counting process. A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.

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EXAMPLES 1.Let x be the number of cars that travel through McDonald’s drive-through in the next hour. 2.Let x be the speed of the next car that passes a state trooper. 3.Let x be the number of As earned in a section of statistics with 15 students enrolled. Determine whether the following are discrete or continuous random variables.

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PROBABILITY HISTORGRAM A probability histogram is like a relative frequency histogram with probabilities instead of relative frequencies.

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EXAMPLES 1.Suppose you toss a coin three times. Let x be the total number of heads. Draw a probability histogram for x. 2.Suppose you throw a pair of dice. Let x be the sum of the numbers on the dice. Draw a probability histogram for x.

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REQUIREMENTS FOR A PROBABILITY DISTRIBUTION ΣP(x) = 1where x assumes all possible values 0 ≤ P(x) ≤ 1for every individual value of x

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EXAMPLES xP(x)P(x) 10.20 20.35 30.12 40.40 5−0.07 xP(x)P(x) 10.20 20.25 30.10 40.14 50.49 Determine if the following are probability distributions xP(x)P(x) 10.20 20.25 30.10 40.14 50.31 (a)(b)(c)

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MEAN, VARIANCE, AND STANDARD DEVIATION Mean of a Prob. Dist. Variance of a Prob. Dist. Standard Deviation of a Prob. Dist.

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1.Enter values for random variable in L 1. 2.Enter the probabilities for the random variables in L 2. 3.Run “1-VarStat L 1, L 2 ” 4.The mean will be. The standard deviation will be σx. To get the variance, square σx. FINDIND MEAN, VARIANCE, AND STANDARD DEVIATION WITH TI-83/84 CALCULATOR

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ROUND-OFF RULE FOR μ, σ, AND σ 2 Round results by carrying one more decimal place than the number of decimal places used for the random variable x.

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MINIMUM AND MAXIMUM USUAL VALUES Recall: minimum usual value = μ − 2σ maximum usual value = μ + 2σ

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EXAMPLE Use the range rule of thumb to determine the unusual values for rolling a pair of dice.

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IDENTIFYING UNUSUAL RESULTS USING PROBABILITIES Rare Event Rule: If, under a given assumption the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. Unusually High: x successes among n trials is an unusually high number of successes if P(x or more) is very small (such as 0.05 or less). Unusually Low: x successes among n trials is an unusually low number of successes if P(x or fewer) is very small (such as 0.05 or less).

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EXAMPLE Consider the procedure of rolling a pair of dice five times and letting x be the number of times that “7” occurs. The table below describes the probability distribution. (a)Find the value of the missing probability. (b)Would it be unusual to roll a pair of dice and get at least three “7s”? xP(x)P(x) 00.402 1 2? 30.032 40.003 50.000+

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EXPECTED VALUE The expected value of a discrete random variable is denoted by E, and it represents the average value of the outcomes. It is obtained by find the value of Σ[x · P(x)]. E = Σ[x · P(x)] EXAMPLE: Exercise 19, 20, page 209.

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