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Chapter 6: Q6.17 (page 272)

Three points X1, X2, X3 are selected at random on a line L. What is the probability that X2 lies between X1 and X3?

Short Answer

Expert verified

The probability that X2 lies between X1 and X3 is13

Step by step solution

01

Content Introduction 

Three points X1 , X2 , X3 are selected at a random on a line L.

Number of ways of arranging and things taking all at a time is n!. Total number of points is 3.

So, the number of ways of arranging 3 points on a line is 3! which is equal to 6.

02

Content Explanation

Possible arrangements are:

X1X2X3X1X3X2X2X1X3X2X3X1X3X1X2X3X1X2

Out of these X2 lies between X1 and X3in two arrangements.

Therefore, Probability that X2lies between X1and X3is=26=13

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Most popular questions from this chapter

If X and Y are jointly continuous with joint density function fX,Y(x, y), show that X + Y is continuous with density function fX+Y(t)=q−qfX,Y(x,t−x)dx

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?

Let X and Y denote the coordinates of a point uniformly chosen in the circle of radius 1centered at the origin. That is, their joint density is f(x,y)=1πx2+y2≤1.

Find the joint density function of the polar coordinates R=(X2+Y2)1/2 and θ=tan-1Y/X.

Repeat Problem 6.3awhen the ball selected is replaced in the urn before the next selection

The random variables XandYhave joint density function f(x,y)=12xy(1−x)0<x<1,0<y<1and equal to0 otherwise.

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(c) FindE[Y].

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