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Chapter 6: Q.6.17 (page 279)

Three points X1,X2,X3 are selected at random on a line L. What is the probability thatX2 lies between X1andX3?

Short Answer

Expert verified

The probability that X2lies between X1andX3is13

Step by step solution

01

To find

The probability thatX2lies between X1andX3

02

Explanation

Three points to consider On a line L,X1,X2,X3are chosen at random.

The number of ways to arrange n items all at once=n!.

Total number of point=3As a result, the number of ways to arrange three points on a line is=3!=6

The possible arrangements are:

X1X2X3X1X3X2X2X1X3X2X3X1X3X1X2X3X2X1

Out of these X2liesbetweenX1andX3in2arrangements

Therefore X2liesbetweenX1andX3=26=13

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Most popular questions from this chapter

The joint probability density function of X and Y is given by

f(x.y)=67(x2+xy2)0<x<1,0<y<2

(a) Verify that this is indeed a joint density function.

(b) Compute the density function of X.

(c) Find P{X > Y}.

(d) Find P{Y > 1 2 |X < 1 2 }.

(e) Find E[X].

(f) Find E[Y].

If X and Y are independent continuous positive random variables, express the density function of (a) Z = X/Y and (b) Z = XY in terms of the density functions of X and Y. Evaluate the density functions in the special case where X and Y are both exponential random variables

If U is uniform on (0,2蟺)and Z, independent of U, is exponential with rate 1, show directly (without using the results of Example 7b) that X and Y defined by

X=2ZcosUY=2ZsinU

are independent standard normal random variables.

The joint density function of X and Y is given by

f(x,y)=xe-x(y+1)x>0,y>0

(a) Find the conditional density of X, given Y = y, and that of Y, given X = x.

(b) Find the density function of Z = XY.

Suppose that X and Y are independent geometric random variables with the same parameter p.

(a) Without any computations, what do you think is the value of P{X = i|X + Y = n}?

Hint: Imagine that you continually flip a coin having probability p of coming up heads. If the second head occurs on the nth flip, what is the probability mass function of the time of the first head?

(b) Verify your conjecture in part (a).
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