Answer all

A longevity study is being conducted on n married hobbit couples. Let p be the probability that an individual hobbit lives at least until his or her eleventy-first birthday, and

assume that the lifespans of di?erent hobbits are independent. Let N0, N1, N2 be the

number of couples in which neither hobbit reaches age eleventy-one, one hobbit does

but not the other, and both hobbits reach eleventy-one, respectively.

(a) Find the joint PMF of N0, N1, N2.

(b) Using (a) and the definition of conditional probability, find the conditional PMF of

N2 given this information, up to a normalizing constant (that is, you do not need to

find the normalizing constant in this part, but just to give a simplified expression that

is proportional to the conditional PMF). For simplicity, you can and should ignore multiplicative constants in this part; this includes multiplicative factors that are functions

of h, since h is now being treated as a known constant.

(c) Now obtain the conditional PMF of N2 using a direct counting argument, now

including any needed normalizing constants so that you are providing a valid conditional

PMF.

(d) Discuss intuitively whether or not p should appear in the answer to (c).

(e) What is the conditional expectation of N2, given the above information (simplify

fully)? This can be done without doing any messy sums, and without having done (b)

or (c).

28. There are n stores in a shopping center, labeled from 1 to n. Let Xi be the number

of customers who visit store i in a particular month, and suppose that X1, X2,...,Xn

are i.i.d. with PMF p(x) = P(Xi = x). Let I ? DUnif(1, 2,...,n) be the label of a

randomly chosen store, so XI is the number of customers at a randomly chosen store.

(a) For i 6= j, find P(Xi = Xj ) in terms of a sum involving the PMF p(x).

(b) Find the joint PMF of I and XI . Are they independent?

(c) Does XI , the number of customers for a random store, have the same marginal

distribution as X1, the number of customers for store 1?

(d) Let J ? DUnif(1, 2,...,n) also be the label of a randomly chosen store, with I and

J independent. Find P(XI = XJ ) in terms of a sum involving the PMF p(x). How does

P(XI = XJ ) compare to P(Xi = Xj ) for fixed i, j with i 6= j?

You are playing an exciting game of Battleship. Your opponent secretly positions ships

on a 10 by 10 grid and you try to guess where the ships are. Each of your guesses is a

hit if there is a ship there and a miss otherwise.

The game has just started and your opponent has 3 ships: a battleship (length 4), a

submarine (length 3), and a destroyer (length 2). (Usually there are 5 ships to start, but

to simplify the calculations we are considering 3 here.) You are playing a variation in

which you unleash a salvo, making 5 simultaneous guesses. Assume that your 5 guesses

are a simple random sample drawn from the 100 grid positions.

Find the mean and variance of the number of distinct ships you will hit in your salvo.

(Give exact answers in terms of binomial coefficient)

Math 211 Math for Business Analysis Homework P2 Probability Density Functions {Taken from \"BriefCaiculus and Mathematics for Business Analysis. '3\" ed. [ASU special edition)" by Waner.} In exercises 1-12, check whether the given function is a probability density function. If a function fails to be a probability density metion, say why. 7. f(x) = i on[1,e] Suppose the following is known about students in their first year of a Mathematics program at university, regarding the introductory Calculus (C), Linear Algebra (L), and Statistics (S) courses . 80% of students take Calculus . 77% of students take Linear Algebra . 20% of students take Statistics . 65% of students take Calculus and Linear Algebra . 5% of students take Linear Algebra and Statistics . 16% of students take Calculus and Statistics . 4% of students take Linear Algebra, Calculus, and Statistics. Suppose we randomly select a citizen in this city. Answer the following questions. 1. What is the probability that this student is studying Calculus or Linear Algebra? 2. What is the probability that, out of these three courses, this student is only studying Statistics? 3. Are C and S independent? Are S and L mutually exclusive? Choose. Choose. More information is needed NO: NO No. Yes Yes; No Next page Yes YesChec 3 Exercise 1-8A (Algo) Allocating product costs between ending inventory and cost of goods sold LO 1-3 3 Stuart Manufacturing Company began operations on January 1. During the year, it started and completed 1720 units of product. The points financial statements are prepared in accordance with GAAP. The company incurred the following costs: Skipped 1. Raw materials purchased and used-$3,080 2. Wages of production workers-$3,490. Book 3. Salaries of administrative and sales personnel-$1,960 4. Depreciation on manufacturing equipment-$5,126. Hint 5. Depreciation on administrative equipment-$1,765. Print References Stuart sold 1,100 units of product. Required a. Determine the total product cost for the year. b. Determine the total cost of the ending Inventory. (Do not round intermediate calculations.) c. Determine the total of cost of goods sold. (Do not round Intermediate calculations.) Total product coat Total cost of ending inventory Total coal of goods sold(X = 3) = 563 ( # 1 = 10 ( 1 ) = 3 Two video products distributors supply video ape boxes to a video production company. Company A sold 100 boxes of which 5 were efective. Company B sold 300 boxes of which 1 were defective. If a box was defective, find e probability that it came from Company B. Bayes' Theorem)