Math 141/166 Summer 2007cHeather Ramsey 1Math 141 - Chapter 8 Lecture NotesThe purpose of this handout is to facilitate your note-taking during lecture. This handout is not a comprehensive summary of the chapter; I will be giving additional notes during lecture which you will also be responsibleto know for exams and quizzes. Several of these examples come from examples and exercises in Finite Mathematics for the Managerial, Life, and Social Sciences, 8th ed. by Tan.Section 8.1 - Distributions of Random VariablesRandom Variable - A random variable is a rule that assigns a number to each outcome of a chance experiment.Example AA coin is tossed three times and the result of each toss is recorded. Let the random variable X denote the number of headsthat occur in the three tosses.(a) What is the sample space of this experiment?(b) Find the value assigned to each outcome of the experiment by the random variable X.(c) Let (X = 2) denote the event that two heads occur. Find the event (X = 2).Finite Discrete Random Variable - A random variable is called finite discrete if it assumes only finitely many values.(The random variable X in Example A is a finite discrete random varialbe.)Infinite Discrete Random Variable - A random variable is said to be infinite discrete if it takes on infinitely many values,which may be arranged in a sequence. (The random variable in Example B below is an infinite discrete random variable.)Math 141/166 Summer 2007cHeather Ramsey 2Example BA coin is tossed repeatedly until a head occurs. Let the random variable Y denote the number of coin tosses in theexperiment. What are the values of Y?Continuous Random Variable - A random variable is called continuous if the values it may assume comprise an intervalof real numbers. (See Example C below.)Example CA cell phone is turned on until the battery runs out. Let the random variable Z denote the length (in hours) of the life ofthe battery. What values may Z assume?Probability Distribution of a Random VariableExample DFind the probability distribution of the random variable associated with the experiment of Example A.Math 141/166 Summer 2007cHeather Ramsey 3Example EA certain club consists of 2 freshmen, 3 sophomores, 1 junior, and 4 seniors. Three members are to be selected to attenda conference. Let the random variable X denote the number of freshmen selected and find the probability distribution forX.Histogram - a graphical representation of a probability distribution of a random variable.Steps for Making a Histogram1. Locate the values of the random variable on the number line.2. Above each value of the random variable, make a rectangle withand.Example FDraw a histogram showing the probability distribution for the number of heads occurring in three coin tosses (fromExample D).Properties of Histograms1. The area of a rectangle associated with a value of a random variable X gives precisely the probability associated withthat value of X.2. The probability associated with more than one value of the random variable X is given by.Math 141/166 Summer 2007cHeather Ramsey 4Histograms on the Calculator1. Enter Data into Lists. Your random variable (X) values should be entered into L1 and the corresponding probabilitiesshould be entered into L2.2. Turn on Stat Plot. Press 2nd Y= and choose which plot you want. In class I used the first one. When on the Stat Plotscreen, highlight the following options:OnHistogram (Bar Graph) picture - the last picture in the top rowXlist: L1Freq: L2NOTE: Once you’ve done this the plot will be turned ON until you turn it OFF.3. Set Window. Use X values to choose Xmin and Xmax. Set Y values between 0 and 1 and choose Yscl by looking atyour data.4. Graph. (Make sure you don’t have anything entered on Y= screen.)Example G(a) Use your calculator to draw a histogram showing the probability distribution given in the following table:Value of XP(X = x)5 .156 .27 .38 .259 .0510 .05(b) Identify the part of the histogram in part (a) whose area gives the probability P(6 ≤ X ≤ 8).Math 141/166 Summer 2007cHeather Ramsey 5Section 8.2 - Expected ValueAverage, or Mean - The average, or mean, of the n numbers x1, x2, . . . ,xnis , whereExample AA certain bank was interested in knowing the average number of cars waiting in line at its drive-in teller, so every 15minutes during one work day, the teller recorded the number of cars waiting in line. Using the information in the tablebelow, find the average number of cars waiting in line at a bank’s drive-in teller on that day.Number of Frequency ofCars Occurrence0 21 92 143 74 4Expected Value of a Random Variable X- Let X denote a random variable that assumes the values x1, x2, . . . ,xnwithassociated probabilities p1, p2, . . . , pn, respectively. Then the expected value of X, written E(X), is given by the followingformula:Math 141/166 Summer 2007cHeather Ramsey 6Example B(#10 pg. 461 Tan)The owner of a newsstand in a college community estimates the weekly demand for a certain magazine as follows:QuantityDemanded 10 11 12 13 14 15Probability .05 .15 .25 .30 .20 .05Find the number of issues of the magazine that the newsstand can expect to sell per week.Note: The expected value of a random variable X can be thought of as.Example CA club is holding a fund-raising raffle. Ten thousand tickets have been sold for $2 each. There will be a first prize of$3000, 3 second prizes of $1000 each, 5 third prizes of $500 each, and 20 consolation prizes of $100 each. Letting Xdenote the net winnings (that is, winnings less the cost of the ticket) associated with the tickets, find E(X). Interpretyour results.Fair Game -Odds in Favor Of and Odds AgainstIf P(E) is the probability of an event E occurring, then1. The odds in favor of E occurring are2. The odds against E occurring areNote: Whenever possible, odds are expressed as. If the odds in favor of E areab, we say the odds in favor of E are. If the odds against E occurring areba, we say the odds against Eare.Math 141/166 Summer 2007cHeather Ramsey 7Example DThe probability of an event E occurring is 0.8. What are the odds in favor of E occurring? What are the odds against Eoccurring?Probability of an Event (Given the Odds)If the odds in favor of an event E occurring are a to b, then the probability of E occurring isP(E) =Example EConsider each of the following statements.a) “The odds that Sonny’s Greased Lightning will win the horse race are 9 to 5.”b) “The odds that it will not rain tomorrow
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