Understanding Basic Statistics
8th Edition
ISBN: 9781337558075
Author: Charles Henry Brase, Corrinne Pellillo Brase
Publisher: Cengage Learning
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Chapter 4, Problem 1UTA
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16. Show that, if X and Y are independent random variables, such that E|X|< ∞,
and B is an arbitrary Borel set, then
EXI{Y B} = EX P(YE B).
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.
Chapter 4 Solutions
Understanding Basic Statistics
Ch. 4.1 - Note: Answers may vary due to rounding....Ch. 4.1 - Note: Answers may vary due to rounding....Ch. 4.1 - Note: Answers may vary due to rounding....Ch. 4.1 - Note: Answers may vary due to rounding....Ch. 4.1 - Critical Thinking: Linear Correlation Look at the...Ch. 4.1 - Critical Thinking: Linear Correlation Look at the...Ch. 4.1 - Critical Thinking: Lurking Variables Over the past...Ch. 4.1 - Critical Thinking: Lurking Variables Over the past...Ch. 4.1 - Critical Thinking: Lurking Variables Over the past...Ch. 4.1 - Critical Thinking: Lurking Variables Over the past...
Ch. 4.1 - Interpretation Trevor conducted a study and found...Ch. 4.1 - Interpretation Do people who spend more time on...Ch. 4.1 - Veterinary Science: Shetland Ponies How much...Ch. 4.1 - Health Insurance:Administrative Cost The following...Ch. 4.1 - Meteorology: Cyclones Can a low barometer reading...Ch. 4.1 - Geology: Earthquakes Is the magnitude of an...Ch. 4.1 - Baseball: Batting Averages and Home Runs In...Ch. 4.1 - University Crime: FBI Report Do larger...Ch. 4.1 - Prob. 19PCh. 4.1 - Prob. 20PCh. 4.1 - Expand Your Knowledge: Using a Table to Test The...Ch. 4.1 - Expand Your Knowledge: Sample Size and...Ch. 4.1 - Prob. 23PCh. 4.2 - Statistical Literacy In the least-squares line...Ch. 4.2 - Statistical Literacy In the least squares line...Ch. 4.2 - Critical Thinking When we use a least-squares line...Ch. 4.2 - Critical Thinking If two variables have a negative...Ch. 4.2 - Critical Thinking: Interpreting Computer Printouts...Ch. 4.2 - Critical Thinking: Interpreting Computer Printouts...Ch. 4.2 - Economics: Entry-Level Jobs An economist is...Ch. 4.2 - Ranching: Cattle You are the foreman of the Bar-S...Ch. 4.2 - Weight of Car: Miles per Gallon Do heavier cars...Ch. 4.2 - Basketball: Fouls Data for this problem are based...Ch. 4.2 - Auto Accidents: Age Data for this problem are...Ch. 4.2 - Auto Accidents: Age Let x be the age of a licensed...Ch. 4.2 - Incoine: Medicai Care Let x be per capita income...Ch. 4.2 - Violent Crimes: Prisons Does prison really deter...Ch. 4.2 - Education: Violent Crime The following data are...Ch. 4.2 - Research: Patents The following data are based on...Ch. 4.2 - Archaeology: Artifacts Data for this problem are...Ch. 4.2 - Cricket Chirps: Temperature Anyone who has been...Ch. 4.2 - Expand Your Knowledge: Residual Plot The...Ch. 4.2 - Residual Plot: Miles per Gallon Consider the data...Ch. 4.2 - Expand Your knowledge: Logarithmic...Ch. 4.2 - Expand Your Knowledge: Logarithmic...Ch. 4.2 - Prob. 24PCh. 4.2 - Expand Your Knowledge: Logarithmic...Ch. 4 - Terminology Consider the equation of a...Ch. 4 - Terminology Consider the values of the sample...Ch. 4 - Terminology Suppose we have a set of ordered pairs...Ch. 4 - Terminology Consider the following terms in a...Ch. 4 - Statistical Literacy Suppose the scatter diagram...Ch. 4 - Critical Thinking Suppose you and a friend each...Ch. 4 - Statistical Literacy When using the least-squares...Ch. 4 - StatisticalLiteracy Suppose that for x = 3. the...Ch. 4 - In Problems 9-14, (a) Draw a scatter diagram for...Ch. 4 - In Problems 9-14, (a) Draw a scatter diagram for...Ch. 4 - In Problems 9-14, (a) Draw a scatter diagram for...Ch. 4 - In Problems 9-14, (a) Draw a scatter diagram for...Ch. 4 - In Problems 9-14, (a) Draw a scatter diagram for...Ch. 4 - In Problems 9-14, (a) Draw a scatter diagram for...Ch. 4 - Prob. 1UTACh. 4 - Prob. 2UTACh. 4 - Prob. 3UTACh. 4 - Prob. 4UTACh. 4 - The data in this section are taken from this...Ch. 4 - The data in this section are taken from this...
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- 8. Show that, if {Xn, n ≥ 1) are independent random variables, then sup X A) < ∞ for some A.arrow_forward8- 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_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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