A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 2, Problem 2.5TE
For any sequence of
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I need some assistance solving Part B of this question. Refer to the excel data in the image provided to answer Part B. SoftBus Company sells PC equipment and customized software to small companies to help them manage their day-to-day business activities. Although SoftBus spends time with all customers to understand their needs, the customers are eventually on their own to use the equipment and software intelligently. To understand its customers better, SoftBus recently sent questionnaires to a large number of prospective customers. Key personnel—those who would be using the software—were asked to fill out the questionnaire. SoftBus received 82 usable responses, as shown in the file. You can assume that these employees represent a random sample of all of SoftBus's prospective customers. SoftBus believes it can afford to spend much less time with customers who own PCs and score at least 4 on PC Knowledge. Let's call these the "PC-savvy" customers. On the other hand, SoftBus believes it…
Suppose you are gambling on a roulette wheel. Each time the wheel is spun, the result is one of the outcomes 0, 1, and so on through 36. Of these outcomes, 18 are red, 18 are black, and 1 is green. On each spin you bet $5 that a red outcome will occur and $1 that the green outcome will occur. If red occurs, you win a net $4. (You win $10 from red and nothing from green.) If green occurs, you win a net $24. (You win $30 from green and nothing from red.) If black occurs, you lose everything you bet for a loss of $6.
a. Use simulation to generate 1,000 plays from this strategy. Each play should indicate the net amount won or lost. Then, based on these outcomes, calculate a 95% confidence interval for the total net amount won or lost from 1,000 plays of the game. (Round your answers to two decimal places and if your answer is negative value, enter "minus" sign.)
Lower Limit
Upper Limit
B1 The x distribution is a special case of Gamma distribution (not to be confused with gamma
function; see below). The density function of the Gamma distribution with parameters
and k is given by
where
-1
e
-x/0
(x) =
=
г(k) Øk
if x > 0, and
otherwise,
г(k) = √ ₁
k-1-x dx
x' e
is the gamma function.
(a) For every k ≥ 1, 0 > 0, find the mode of the density. Hint: The algebra can be
simplified by appropriate use of logarithms.
~
Now suppose that X1,..., Xn id Exp(\) and that we have a prior belief in A which is
consistent with a prior distribution X. Gamma(a, b), for some a, ß, i.e. the prior density
of is
Baxa-1-BA
T(a)
e
=
so 01/ẞ and k = a.
(b) Write down the likelihood, and show that the posterior distribution for \ is also a
Gamma distribution, but with parameters a +n and B + Σ Xi.
(c) Find the mode of the posterior distribution and examine the behaviour as n → ∞.
Chapter 2 Solutions
A First Course in Probability (10th Edition)
Ch. 2 - A box contains 3 marbles: 1 red, 1 green, and 1...Ch. 2 - In an experiment, die is rolled continually until...Ch. 2 - Two dice are thrown. Let E be the event that the...Ch. 2 - A, B, and C take turns flipping a coin. The first...Ch. 2 - A system is composed of 5 components, each of...Ch. 2 - A hospital administrator codes incoming patients...Ch. 2 - Consider an experiment that consists of...Ch. 2 - Suppose that A and B are mutually exclusive events...Ch. 2 - A retail establishment accepts either the American...Ch. 2 - Sixty percent of the students at a certain school...
Ch. 2 - A total of 28 percent of American males smoke...Ch. 2 - An elementary school is offering 3 language...Ch. 2 - A certain town with a population of 100.000 has 3...Ch. 2 - The following data were given in a study of a...Ch. 2 - If it is assumed that all (525) poker hands are...Ch. 2 - Poker dice is played by simultaneously rolling 5...Ch. 2 - Twenty five people, consisting of 15 women and 10...Ch. 2 - Two cards are randomly selected from an ordinary...Ch. 2 - Two symmetric dice have had two of their sides...Ch. 2 - Suppose that you are playing blackjack against a...Ch. 2 - A small community organization consists of 20...Ch. 2 - Consider the following technique for shuffling a...Ch. 2 - A pair of fair dice is rolled. What is the...Ch. 2 - It two dice are rolled, what is the probability...Ch. 2 - A pair of dice is rolled until a sum of either 5...Ch. 2 - The game of craps is played as follows: A player...Ch. 2 - An urn contains 3 red and 7 black balls. Players A...Ch. 2 - An urn contains 5 red, 6 blue, and 8 green balls....Ch. 2 - An urn contains n white and m black balls, where n...Ch. 2 - The chess clubs of two schools consist of,...Ch. 2 - A 3-person basketball team consists of a guard, a...Ch. 2 - A group of individuals containing b boys and g...Ch. 2 - A forest contains 20 elk, of which 5 are captured,...Ch. 2 - The second Earl of Yarborough is reported to have...Ch. 2 - Seven balls are randomly withdrawn from an urn...Ch. 2 - Two cards are chosen at random from a deck of 52...Ch. 2 - An instructor gives her class a set of 10 problems...Ch. 2 - There are n socks. 3 of which are red, in a...Ch. 2 - There are 5 hotels in a certain town. If 3 people...Ch. 2 - If 4 balls are randomly chosen from an urn...Ch. 2 - If a die is rolled 4 times, what is the...Ch. 2 - Two dice are thrown n times in succession. Compute...Ch. 2 - a. If N people, including A and B, are randomly...Ch. 2 - Five people, designated as A, B, C, D, E, are...Ch. 2 - A woman has n keys, of which one will open her...Ch. 2 - How many people have to be in a room in order that...Ch. 2 - Suppose that 5 of the numbers 1, 2,..., 14 are...Ch. 2 - Given 20 people, what is the probability that...Ch. 2 - A group of 6 men and 6 women is randomly divided...Ch. 2 - In a hand of bridge, find the probability that you...Ch. 2 - Suppose that n balls are randomly distributed into...Ch. 2 - A closet contains 10 pairs of shoes. If 8 shoes...Ch. 2 - If 8 people, consisting of 4 couples, are randomly...Ch. 2 - Compute the probability that a bridge hand is void...Ch. 2 - Compute the probability that a hand of 13 cards...Ch. 2 - Two players play the following game: Player A...Ch. 2 - Prove the following relations: EFEEFCh. 2 - Prove the following relations: If EF, then FCEC.Ch. 2 - Prove the following relations: 3. F=FEFEC and...Ch. 2 - Prove the following relations: (1Ei)F=1EiF and...Ch. 2 - For any sequence of events E1,E2,..., define a new...Ch. 2 - Let E, F, and C be three events. Find expressions...Ch. 2 - Use Venn diagrams a. to simplify the expression...Ch. 2 - Prob. 2.8TECh. 2 - Suppose that an experiment is performed n times...Ch. 2 - Prove...Ch. 2 - If P(E)=.9 and P(F)=.8, show that P(EF).7. In...Ch. 2 - Show that the probability that exactly one of the...Ch. 2 - Prove that P(EF)=P(E)P(EF).Ch. 2 - Prove Proposition 4.4 by mathematical induction.Ch. 2 - An urn contains M white and N black balls. If a...Ch. 2 - Use induction to generalize Bonferronis inequality...Ch. 2 - Consider the matching problem. Example 5m, and...Ch. 2 - Let fn, denote the number of ways of tossing a...Ch. 2 - An urn contains n red and m blue balls. They are...Ch. 2 - Consider an experiment whose sample space consists...Ch. 2 - Consider Example 50, which is concerned with the...Ch. 2 - A cafeteria offers a three-course meal consisting...Ch. 2 - A customer visiting the suit department of a...Ch. 2 - A deck of cards is dealt out. What is the...Ch. 2 - Let A denote the event that the midtown...Ch. 2 - An ordinary deck of 52 cards is shuffled. What is...Ch. 2 - Urn A contains 3 red and 3 black balls, whereas...Ch. 2 - In a state lottery, a player must choose 8 of the...Ch. 2 - From a group of 3 first-year students, 4...Ch. 2 - For a finite set A, let N(A) denote the number of...Ch. 2 - Consider an experiment that consists of 6 horses,...Ch. 2 - A 5-card hand is dealt from a well-shuffled deck...Ch. 2 - A basketball team consists of 6 frontcourt and 4...Ch. 2 - Suppose that a person chooses a letter at random...Ch. 2 - Prove Booles inequality P(i=1Ai)i=1P(Ai)Ch. 2 - Show that if P(Ai)=1 for all i1, then P(i=1Ai)=1.Ch. 2 - Let Tk(n) denote the number of partitions of the...Ch. 2 - Five balls are randomly chosen, without...Ch. 2 - Four red, 8 blue, and 5 green balls are randomly...Ch. 2 - Ten cards are randomly chosen from a deck of 52...Ch. 2 - Balls are randomly removed from an urn initially...
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