Essentials of Statistics, Books a la Carte Edition (5th Edition)
5th Edition
ISBN: 9780321926739
Author: Mario F. Triola
Publisher: PEARSON
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Chapter 2.3, Problem 1BSC
To determine
To explain: The advantage for examining the histogram instead of the frequency distribution.
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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 2 Solutions
Essentials of Statistics, Books a la Carte Edition (5th Edition)
Ch. 2.2 - Frequency Distribution Table 2-2 on page 45 is a...Ch. 2.2 - Relative Frequency Distribution After construction...Ch. 2.2 - Do You Believe? In a Harris Interactive survey,...Ch. 2.2 - Analyzing a Frequency Distribution The...Ch. 2.2 - In Exercises 5-10, identify the class width, class...Ch. 2.2 - In Exercises 5-10, identify the class width, class...Ch. 2.2 - In Exercises 5-10, identify the class width, class...Ch. 2.2 - In Exercises 5-10, identify the class width, class...Ch. 2.2 - In Exercises 5-10, identify the class width, class...Ch. 2.2 - In Exercises 5-10, identify the class width, class...
Ch. 2.2 - Normal Distributions. In Exercises 11-14, answer...Ch. 2.2 - Normal Distributions. In Exercises 11-14, answer...Ch. 2.2 - Normal Distributions. In Exercises 11-14, answer...Ch. 2.2 - Normal Distributions. In Exercises 11-14, answer...Ch. 2.2 - Relative Frequencies for Comparisons. In Exercises...Ch. 2.2 - Relative Frequencies for Comparisons. In Exercises...Ch. 2.2 - Cumulative Frequency Distributions. In Exercises...Ch. 2.2 - Cumulative Frequency Distributions. In Exercises...Ch. 2.2 - Analysis of Last Digits Heights of statistics...Ch. 2.2 - Analysis of Last Digits Weights of respondents...Ch. 2.2 - Constructing Frequency Distributions. In Exercises...Ch. 2.2 - Constructing Frequency Distributions. In Exercises...Ch. 2.2 - Constructing Frequency Distributions. In Exercises...Ch. 2.2 - Constructing Frequency Distributions. In Exercises...Ch. 2.2 - Constructing Frequency Distributions. In Exercises...Ch. 2.2 - Constructing Frequency Distributions. In Exercises...Ch. 2.2 - Constructing Frequency Distributions. In Exercises...Ch. 2.2 - Constructing Frequency Distributions. In Exercises...Ch. 2.2 - Categorical Data. In Exercises 29-32, use the...Ch. 2.2 - Categorical Data. In Exercises 29-32, use the...Ch. 2.2 - Categorical Data. In Exercises 29-32, use the...Ch. 2.2 - Categorical Data. In Exercises 29-32, use the...Ch. 2.2 - Interpreting Effects of Outliers Refer to Data Set...Ch. 2.2 - Number of Classes According to what is known as...Ch. 2.3 - Prob. 1BSCCh. 2.3 - Voluntary Response Sample The histogram in Figure...Ch. 2.3 - Small Data NASA provides these duration times (in...Ch. 2.3 - Normal Distribution When it refers to a normal...Ch. 2.3 - Interpreting a Histogram. In Exercises 5-8, answer...Ch. 2.3 - Interpreting a Histogram. In Exercises 5-8, answer...Ch. 2.3 - Interpreting a Histogram. In Exercises 5-8, answer...Ch. 2.3 - Interpreting a Histogram. In Exercises 5-8, answer...Ch. 2.3 - Analysis of Last Digits Use the frequency...Ch. 2.3 - Analysis of Last Digits Use the frequency...Ch. 2.3 - Constructing Histograms. In Exercises 9-18,...Ch. 2.3 - Constructing Histograms. In Exercises 9-18,...Ch. 2.3 - Constructing Histograms. In Exercises 9-18,...Ch. 2.3 - Constructing Histograms. In Exercises 9-18,...Ch. 2.3 - Prob. 15BSCCh. 2.3 - Constructing Histograms. In Exercises 9-18,...Ch. 2.3 - Constructing Histograms. In Exercises 9-18,...Ch. 2.3 - Constructing Histograms. In Exercises 9-18,...Ch. 2.3 - Back-to-Back Relative Frequency Histograms When...Ch. 2.3 - Interpreting a Histogram. In Exercises 5-8, answer...Ch. 2.4 - Bar Chart and Pareto Chart A bar chart and a...Ch. 2.4 - Scatterplot What is a scatterplot? What type of...Ch. 2.4 - SAT Scores Listed below are SAT scores from a...Ch. 2.4 - SAT Scores Given that the data in Exercise 3 were...Ch. 2.4 - Scatterplots. In Exercises 5-8, use the given...Ch. 2.4 - Scatterplots. In Exercises 5-8, use the given...Ch. 2.4 - Scatterplots. In Exercises 5-8, use the given...Ch. 2.4 - Scatterplots. In Exercises 5-8, use the given...Ch. 2.4 - Time-Series Graphs. In Exercises 9 and 10,...Ch. 2.4 - Time-Series Graphs. In Exercises 9 and 10,...Ch. 2.4 - Dotplots. In Exercises II and 12, construct the...Ch. 2.4 - Dotplots. In Exercises 11 and 12, construct the...Ch. 2.4 - Stemplots. In Exercises 13 and 14, construct the...Ch. 2.4 - Stemplots. In Exercises 13 and 14, construct the...Ch. 2.4 - Pareto Charts. In Exercises 15 and 16, construct...Ch. 2.4 - Pareto Charts. In Exercises 15 and 16, construct...Ch. 2.4 - Pie Charts. In Exercises 17 and 18, construct the...Ch. 2.4 - Pie Charts. In Exercises 17 and 18, construct the...Ch. 2.4 - Frequency Polygon. In Exercises 19 and 20,...Ch. 2.4 - Frequency Polygon. In Exercises 19 and 20,...Ch. 2.4 - Deceptive Graphs. In Exercises 21-24, identify the...Ch. 2.4 - Deceptive Graphs. In Exercises 21-24, identify the...Ch. 2.4 - Deceptive Graphs. In Exercises 21-24, identify the...Ch. 2.4 - Deceptive Graphs. In Exercises 21-24, identify the...Ch. 2.4 - Back-to-Back Stemplots Exercise 19 in Section 2-3...Ch. 2 - When one is constructing a table representing the...Ch. 2 - When one is constructing a table representing the...Ch. 2 - When one is constructing a table representing the...Ch. 2 - A stemplot is created from the intervals (min)...Ch. 2 - In the California Daily 4 lottery, four digits...Ch. 2 - In an investigation of the travel costs of college...Ch. 2 - In an investigation of the relationship between...Ch. 2 - As a quality control manager at Sony, you find...Ch. 2 - What characteristic of a data set can be better...Ch. 2 - A histogram is to be constructed from the brain...Ch. 2 - Frequency Distribution of Brain Volumes Construct...Ch. 2 - Histogram of Brain Volumes Construct the histogram...Ch. 2 - Dotplot of California Lottery In the California...Ch. 2 - Stemplot of IQ Scores Listed below are the first...Ch. 2 - CO Emissions Listed below are the amounts (million...Ch. 2 - CO and NO Emissions Exercise 5 lists the amounts...Ch. 2 - Sports Equipment According to USA Today, the...Ch. 2 - In Exercises 1-5, refer to the table in the...Ch. 2 - In Exercises 1-5, refer to the table in the...Ch. 2 - In Exercises 1-5, refer to the table in the...Ch. 2 - In Exercises 1-5, refer to the table in the...Ch. 2 - In Exercises 1-5, refer to the table in the...Ch. 2 - Grooming Time Listed below are times (minutes)...Ch. 2 - Histogram of Grooming Times Use the frequency...Ch. 2 - Stemplot of Grooming Times Use the data from...Ch. 2 - Technology Project It was noted in this section...Ch. 2 - Flight Planning Data Set 15 in Appendix B includes...Ch. 2 - Prob. 2FDDCh. 2 - Flight Planning Data Set 15 in Appendix B includes...
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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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