Homework Set Module 9

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Apr 3, 2024

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Gloria Hansen Homework Set – Module 9 4.1) Which of the following describe con5nuous random variables? Which describe discrete random variables? a) The number of newspapers sold by the New York Times each month Discrete b) The amount of ink used in prin5ng a Sunday edi5on of the New York Times Con5nuous c) The actual number of ounces in a 1-gallon boFle of laundry detergent Con5nuous d) The number of defec5ve parts in a shipment of nuts and bolts Discrete e) The number of people collec5ng unemployment insurance each month Discrete 4.2) Security analysts are professionals who devote full-5me efforts to evalua5ng the investment worth of a narrow list of stocks. The following variables are of interest to security analysts. Which are discrete and which are con5nuous random variables? a) The closing price of a par5cular stock on the New York Stock exchange Con5nuous b) The number of shares of a par5cular stock that are traded each business day Discrete c) The quarterly earnings of a par5cular firm Con5nuous d) The percentage change in earnings between last year and this year for a par5cular firm Con5nuous e) The number of new products introduced per year by a firm. Discrete f) The 5me un5l a pharmaceu5cal company gains approval from the US Food and Drug Administra5on to market a new drug Con5nuous 4.12) The random variable x has the following discrete probability distribu5on: a) List the values x may assume. 1, 3, 5, 7, 9 b) What value of x is most probable? 5 c) Display the probability distribu5on as a graph.
Gloria Hansen d) Find p(x = 7) .2 e) Find p(x ³ 5) (.4 + .2 +.1) = .7 f) Find p(x > 2) (.2 +.4 + .2 + .1) = .9 g) Find E(x) (1*.1)+(3*.2)+(5*.4)+(7*.2)+(9*.1) = 5 4.14) Explain why each of the following is or is not a valid probability distribu5on for a discrete random variable x: a) P(x) does not equal 1 b) Valid probability distribu5on c) Nega5ve p(x) value d) P(x) does not equal 1 4.20) Refer to the PC Magazine (December 2019) ar5cle on parents’ opinions on the appropriate age a child should receive their first digital device. Let x represent the age (in years) a parent would buy their child their first smart speaker (e.g., Amazon Echo, Google Home). Consider the probability distribu5on for x (adapted from the ar5cle) shown in the table. a) Verify that the probabili5es in the table sum to 1. S P(x) = Valid b) Find P(x < 6) .03+.04.+.05+.08+.07+.07+.06+.09+0.5+.15+.15+.09 = .93 c) Find P(10 < x < 15) .08+.07+.07+.06+.09+.05 = .42 d) Find E(x) and prac5cally interpret the value. S P(x)x = 13.08 The average age that they think is appropriate to buy a child their first speaker is around 13 years old. 4.35) The Na5onal Weather Service issues precipita5on forecasts that indicate the likelihood of measurable precipita5on (≥ .01 inch) at a specific point (the official rain gauge) during a given
Gloria Hansen 5me period. Suppose that if a measurable amount of rain falls during the next 24 hours, a river will reach flood stage and a business will incur damages of $300,000. The Na5onal Weather Service has indicated that there is a 30% chance of a measurable amount of rain during the next 24 hours. a) Construct the probability distribu5on that describes the poten5al flood damages. P(0) = .70 P(300,000) = .30 b) Find the firm’s expected loss die to flood damage. 300,000(.30) = 90,000 4.41) Consider the following probability distribu5on: P(x) = (5, x)(.7) x (.3) 5-x (x = 0, 1, 2, …, 5) a) Is x a discrete or con5nuous random variable? Discrete random variable b) What is the name of this probability distribu5on? X follows a binomial distribu5on (X ~ binomial (5, .7)) c) Graph the probability distribu5on. d) Find the mean and standard devia5on of x. Mean = 5 * .7 = 3.5 Standard devia5on = sqrt(5 * .7 * (1-.7)) = 1.0247 e) Show the mean and the 2-standard-devia5on interval on each side of the mean on the graph you drew in part c.
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