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Chapter 6: Q.6.11 (page 278)

Let X1,X2,...be a sequence of independent uniform (0,1)random variables. For a fixed constant c, define the random variable N by N=min{n:Xn>c}Is N independent ofXN? That is, does knowing the value of the first random variable that is greater than c affect the probability distribution of when this random variable occurs? Give an intuitive explanation for your answer.

Short Answer

Expert verified

Yes, NisindependentbyXn.

Step by step solution

01

Content Introduction

A random variable is a variable with an unknown value or a function that gives values to each of the results of an experiment. It's possible for a random variable to be discrete or continuous.

02

Explanation

Yes,NisindependentbyXn.

This is simply demonstrated by considering the Equivalent question of whetherXnis independent of N. However, this is correct, because knowing when the first random variable higher thanCarises has no bearing on the probability distribution of its value, which is the uniform distribution on (C,1).

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Most popular questions from this chapter

An insurance company supposes that each person has an accident parameter and that the yearly number of accidents of someone whose accident parameter is 位 is Poisson distributed with mean 位. They also suppose that the parameter value of a newly insured person can be assumed to be the value of a gamma random variable with parameters s and 伪. If a newly insured person has n accidents in her first year, find the conditional density of her accident parameter. Also, determine the expected number of accidents that she will have in the following year.

Let X1, X2, X3, X4, X5 be independent continuous random variables having a common distribution function F and density function f, and set I = P{X1 < X2 < X3 < X4 < X5}

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(b) Evaluate I.

(c) Give an intuitive explanation for your answer to (b).

Suppose that A, B, C, are independent random variables, each being uniformly distributed over0,1.

(a) What is the joint cumulative distribution function of A, B, C?

(b) What is the probability that all of the roots of the equation AX2+Bx+C=0are real?

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)=1蟺x2+y2鈮�1.

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

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

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