Consider a directory of classified advertisements that consists of m pages, where m is very large. Suppose that the number of advertisements per page varies and that your only method of finding out how many advertisements there are on a specified page is to count them. In addition, suppose that there are too many pages for it to be feasible to make a complete count of the total number of advertisements and that your objective is to choose a directory advertisement in such a way that each of them has an equal chance of being selected. a. If you randomly choose a page and then randomly choose an advertisement from that page, would that satisfy your objective? Why or why not? Let n ( i ) denote the number of advertisements on page i . i = 1. . . .. m , and suppose that whereas these quantities are unknown, we can assume that they are all less than or equal to some specified value n. Consider the following algorithm for choosing an advertisement. Step 1. Choose a page at random. Suppose it is page X. Determine n(X) by counting the number of advertisements on page X. Step 2. “Accept” page X with probability n ( x ) n . If page X is accepted, go to step 3. Otherwise, return to step 1. Step 3. Randomly choose one of the advertisements on page X. Call each pass of the algorithm through step 1 an iteration. For instance, if the first randomly chosen page is rejected and the second accepted, then we would have needed 2 iterations of the algorithm to obtain an advertisement. b. What is the probability that a single iteration of the algorithm results in the acceptance of an advertisement on page i? c. What is the probability that a single iteration of the algorithm results in the acceptance of an advertisement? d. What is the probability that the algorithm goes through k iterations, accepting the jth advertisement on page i on the final iteration? e. What is the probability that the jth advertisement on page i is the advertisement obtained from the algorithm? f. What is the expected number of iterations taken by the algorithm?
Consider a directory of classified advertisements that consists of m pages, where m is very large. Suppose that the number of advertisements per page varies and that your only method of finding out how many advertisements there are on a specified page is to count them. In addition, suppose that there are too many pages for it to be feasible to make a complete count of the total number of advertisements and that your objective is to choose a directory advertisement in such a way that each of them has an equal chance of being selected. a. If you randomly choose a page and then randomly choose an advertisement from that page, would that satisfy your objective? Why or why not? Let n ( i ) denote the number of advertisements on page i . i = 1. . . .. m , and suppose that whereas these quantities are unknown, we can assume that they are all less than or equal to some specified value n. Consider the following algorithm for choosing an advertisement. Step 1. Choose a page at random. Suppose it is page X. Determine n(X) by counting the number of advertisements on page X. Step 2. “Accept” page X with probability n ( x ) n . If page X is accepted, go to step 3. Otherwise, return to step 1. Step 3. Randomly choose one of the advertisements on page X. Call each pass of the algorithm through step 1 an iteration. For instance, if the first randomly chosen page is rejected and the second accepted, then we would have needed 2 iterations of the algorithm to obtain an advertisement. b. What is the probability that a single iteration of the algorithm results in the acceptance of an advertisement on page i? c. What is the probability that a single iteration of the algorithm results in the acceptance of an advertisement? d. What is the probability that the algorithm goes through k iterations, accepting the jth advertisement on page i on the final iteration? e. What is the probability that the jth advertisement on page i is the advertisement obtained from the algorithm? f. What is the expected number of iterations taken by the algorithm?
Solution Summary: The author analyzes the probability that a single iteration of the algorithm results in the acceptance of an advertisement on page.
Consider a directory of classified advertisements that consists of m pages, where m is very large. Suppose that the number of advertisements per page varies and that your only method of finding out how many advertisements there are on a specified page is to count them. In addition, suppose that there are too many pages for it to be feasible to make a complete count of the total number of advertisements and that your objective is to choose a directory advertisement in such a way that each of them has an equal chance of being selected.
a. If you randomly choose a page and then randomly choose an advertisement from that page, would that satisfy your objective? Why or why not?
Let
n
(
i
)
denote the number of advertisements on page
i
.
i
=
1.
.
.
..
m
, and suppose that whereas these quantities are unknown, we can assume that they are all less than or equal to some specified value n. Consider the following algorithm for choosing an advertisement.
Step 1. Choose a page at random. Suppose it is page X. Determine n(X) by counting the number of advertisements on page X.
Step 2. “Accept” page X with probability
n
(
x
)
n
. If page X is accepted, go to step 3. Otherwise, return to step 1.
Step 3. Randomly choose one of the advertisements on page X.
Call each pass of the algorithm through step 1 an iteration. For instance, if the first randomly chosen page is rejected and the second accepted, then we would have needed 2 iterations of the algorithm to obtain an advertisement.
b. What is the probability that a single iteration of the algorithm results in the acceptance of an advertisement on page i?
c. What is the probability that a single iteration of the algorithm results in the acceptance of an advertisement?
d. What is the probability that the algorithm goes through k iterations, accepting the jth advertisement on page i on the final iteration?
e. What is the probability that the jth advertisement on page i is the advertisement obtained from the algorithm?
f. What is the expected number of iterations taken by the algorithm?
ball is drawn from one of three urns depending on the outcomeof a roll of a dice. If the dice shows a 1, a ball is drawn from Urn I, whichcontains 2 black balls and 3 white balls. If the dice shows a 2 or 3, a ballis drawn from Urn II, which contains 1 black ball and 3 white balls. Ifthe dice shows a 4, 5, or 6, a ball is drawn from Urn III, which contains1 black ball and 2 white balls. (i) What is the probability to draw a black ball? [7 Marks]Hint. Use the partition rule.(ii) Assume that a black ball is drawn. What is the probabilitythat it came from Urn I? [4 Marks]Total marks 11 Hint. Use Bayes’ rule
Let X be a random variable taking values in (0,∞) with proba-bility density functionfX(u) = 5e^−5u, u > 0.Let Y = X2 Total marks 8 . Find the probability density function of Y .
Let P be the standard normal distribution, i.e., P is the proba-bility measure on R, B(R) given bydP(x) = 1√2πe− x2/2dx.Consider the random variablesfn(x) = (1 + x2) 1/ne^(x^2/n+2) x ∈ R, n ∈ N.Using the dominated convergence theorem, prove that the limitlimn→∞E(fn)exists and find it
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