Introductory Statistics (2nd Edition)
2nd Edition
ISBN: 9780321978271
Author: Robert Gould, Colleen N. Ryan
Publisher: PEARSON
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Chapter 3, Problem 92CRE
Children’s Heights Mrs. Diaz has two children: a 3-year-old boy 43 inches tall and a 10-year-old girl 57 inches tall. Three-year-old boys have a
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Proposition 1.1 Suppose that X1, X2,... are random variables. The following
quantities are random variables:
(a) max{X1, X2) and min(X1, X2);
(b) sup, Xn and inf, Xn;
(c) lim sup∞ X
and lim inf∞ Xn-
(d) If Xn(w) converges for (almost) every w as n→ ∞, then lim-
random variable.
→ Xn is a
Exercise 4.2 Prove that, if A and B are independent, then so are A and B, Ac and
B, and A and B.
8. Show that, if {Xn, n ≥ 1) are independent random variables, then
sup X A) < ∞ for some A.
Chapter 3 Solutions
Introductory Statistics (2nd Edition)
Ch. 3 - Earnings A sociologist says, “Typically, men in...Ch. 3 - Houses A real estate agent claims that all things...Ch. 3 - Age of CEOs (Example) the histogram shows the ages...Ch. 3 - Televisions The histogram shows the number of...Ch. 3 - Billionaires According to Forbes.com, the numbers...Ch. 3 - Billionaires According to Forbes.com, the numbers...Ch. 3 - Paid Vacation Days (Example 2) This list...Ch. 3 - Children of First Ladies This list represents the...Ch. 3 - Ages of Presidents at Inauguration At their...Ch. 3 - Ages of Chief Justices at Installation At their...
Ch. 3 - Weight Loss (Example 3) The table shows Minitab...Ch. 3 - Education of Father and Mother The table shows...Ch. 3 - Surfing College students and surfers Rex Robinson...Ch. 3 - Eating Out College student Jacqueline Loya asked a...Ch. 3 - Real state price (Example) look at the two...Ch. 3 - Dice The histogram contain data with a range of 1...Ch. 3 - Birth Weights (Example 5) The mean birth weigh for...Ch. 3 - Birth Length The mean birth length for U.S....Ch. 3 - Children’s Ages (Example 6) Mrs. Johnson’s...Ch. 3 - Pay Rate in Different Currencies The pay rates for...Ch. 3 - Olympics In the most recent summer Olympics, do...Ch. 3 - Weights Suppose you have a data set with the...Ch. 3 - Brain Size The brain size (in hundreds of...Ch. 3 - Happiness A survey on StatCrunch asked people to...Ch. 3 - Drinkers The number of alcoholic drinks per week...Ch. 3 - Prob. 26SECh. 3 - Violent Crime: West (Example 7) In 2011, the mean...Ch. 3 - Violent Crime: East In 2011, the mean rate of...Ch. 3 - Property Crime (Example 8) In 2011, the mean...Ch. 3 - Property Crime In 2011, the mean property crime...Ch. 3 - Heights and z-Scores The dotplot shows heights of...Ch. 3 - Heights Refer to the dotplot in the previous...Ch. 3 - Unusual IQs (Example 9) Wechsler IQ tests have a...Ch. 3 - Lengths of Pregnancy Distributions of gestation...Ch. 3 - Low-Birth-Weight Babies (Example 10) Babies born...Ch. 3 - Birth Lengths Babies born after 40 weeks gestation...Ch. 3 - Women's Heights Assume that women's heights have a...Ch. 3 - SATs The quantitative portion of the SAT exam has...Ch. 3 - Name two measures of the center of a distribution,...Ch. 3 - Name two measures of the variation of a...Ch. 3 - Pixar Animated Movies (Example 11) The ten...Ch. 3 - DreamWorks Animated Movies The ten top-grossing...Ch. 3 - Pixar Animated Movies Again (Example 12) Find the...Ch. 3 - Dreamworks Animated Movies Find the median and...Ch. 3 - Prob. 45SECh. 3 - Drinks The number of alcoholic drinks per week is...Ch. 3 - Outliers a. In your own words, describe to someone...Ch. 3 - Center and Variation When you are comparing two...Ch. 3 - An Error A dieter recorded the number of calories...Ch. 3 - Baseball Strike In 1994, Major League Baseball...Ch. 3 - Heads The graphs show the circumferences of heads...Ch. 3 - House Prices The graphs show the house prices (in...Ch. 3 - Shoes (Example 14) The histograms show the number...Ch. 3 - Tax Rate A StatCrunch survey asked people what...Ch. 3 - Regional Population Density The figure shows the...Ch. 3 - Property Crime Rates The boxplot shows the...Ch. 3 - City Temperatures The boxplot shows temperatures...Ch. 3 - Brain Size The boxplots show the brain size (in...Ch. 3 - Matching Boxplots and Histograms a. Report the...Ch. 3 - Matching Boxplots and Histograms Match each of the...Ch. 3 - Sleep Time of Animals Data at this text's website...Ch. 3 - BA Percentage The data show the percentage of...Ch. 3 - Tall Buildings The dotplot shows the distribution...Ch. 3 - Passing the Bar Exam The dotplot shows the...Ch. 3 - Exam Scores The five-number summary for a...Ch. 3 - Exam Scores The five-number summary for a...Ch. 3 - Death Row: South (Example 15) The table shows the...Ch. 3 - Death Row: West The table shows the numbers of...Ch. 3 - Head Circumference (Example 16) Following are head...Ch. 3 - Heights of Sons and Dads The data at this text’s...Ch. 3 - Final Exam Grades The data that follow are final...Ch. 3 - Speeding Tickets College students Diane Glover and...Ch. 3 - Heights The following graph shows the heights for...Ch. 3 - Marathon Times The following histogram of marathon...Ch. 3 - Soda Consumption A StatCrunch survey asked people...Ch. 3 - Holiday Spending A StatCrunch survey asked people...Ch. 3 - a. State an approximate value for the mean height...Ch. 3 - Ideal Family In 2012, the General Social Survey...Ch. 3 - For exercises 3.85 through 3.88, construct two...Ch. 3 - For exercises 3.85 through 3.88, construct two...Ch. 3 - For exercises 3.85 through 3.88, construct two...Ch. 3 - 3.79-3.82 construct two sets of numbers with at...Ch. 3 - Population Density Data were recorded for each of...Ch. 3 - Population Increase Data were recorded for each of...Ch. 3 - Prob. 85CRECh. 3 - Eating Out, Again College student Jacqueline Loya...Ch. 3 - Study Hours A group of 50 statistics students, 25...Ch. 3 - Driving Accidents College student Sandy Hudson...Ch. 3 - Exam Scores An exam has a mean of 70 and a...Ch. 3 - Boys’ Heights Three-year-old boys in the United...Ch. 3 - SAT and ACT Scores Quantitative SAT scores have a...Ch. 3 - Children’s Heights Mrs. Diaz has two children: a...Ch. 3 - Students’ Ages Here are the ages of some students...Ch. 3 - House Prices The figure, which is from data taken...
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- 8- 6. Show that, for any random variable, X, and a > 0, 8 心 P(xarrow_forward15. This problem extends Problem 20.6. Let X, Y be random variables with finite mean. Show that 00 (P(X ≤ x ≤ Y) - P(X ≤ x ≤ X))dx = E Y — E X.arrow_forward(b) Define a simple random variable. Provide an example.arrow_forward17. (a) Define the distribution of a random variable X. (b) Define the distribution function of a random variable X. (c) State the properties of a distribution function. (d) Explain the difference between the distribution and the distribution function of X.arrow_forward16. (a) Show that IA(w) is a random variable if and only if A E Farrow_forward15. Let 2 {1, 2,..., 6} and Fo({1, 2, 3, 4), (3, 4, 5, 6}). (a) Is the function X (w) = 21(3, 4) (w)+711.2,5,6) (w) a random variable? Explain. (b) Provide a function from 2 to R that is not a random variable with respect to (N, F). (c) Write the distribution of X. (d) Write and plot the distribution function of X.arrow_forward20. Define the o-field R2. Explain its relation to the o-field R.arrow_forward7. Show that An → A as n→∞ I{An} - → I{A} as n→ ∞.arrow_forward7. (a) Show that if A,, is an increasing sequence of measurable sets with limit A = Un An, then P(A) is an increasing sequence converging to P(A). (b) Repeat the same for a decreasing sequence. (c) Show that the following inequalities hold: P (lim inf An) lim inf P(A) ≤ lim sup P(A) ≤ P(lim sup A). (d) Using the above inequalities, show that if A, A, then P(A) + P(A).arrow_forward19. (a) Define the joint distribution and joint distribution function of a bivariate ran- dom variable. (b) Define its marginal distributions and marginal distribution functions. (c) Explain how to compute the marginal distribution functions from the joint distribution function.arrow_forward18. Define a bivariate random variable. Provide an example.arrow_forward6. (a) Let (, F, P) be a probability space. Explain when a subset of ?? is measurable and why. (b) Define a probability measure. (c) Using the probability axioms, show that if AC B, then P(A) < P(B). (d) Show that P(AUB) + P(A) + P(B) in general. Write down and prove the formula for the probability of the union of two sets.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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