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Chapter 6: Q.6.18 (page 272)

Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points X and Y are independent random variables such that X is uniformly distributed over (0, L/2) and Y is uniformly distributed over (L/2, L).] Find the probability that the distance between the two points is greater than L/3

Short Answer

Expert verified

The required probability is79

Step by step solution

01

Content Introduction

In an equation, a variable is a symbol that represents an unknown numerical value.

02

Explanation

We are given that X~Unif(0,L/2). We are required to find P(Y-X>L/3). Firstly, we will construct joint pdf . Since we have chosen points arbitrary, we have that

f(x,y)=fX(x)fY(y)=1(L/2)2=4L2

For (x,y)鈭�(0,L/2)脳(L/2,L)hence the required probability is

localid="1647352265833" P(Y-X>L3)=1-P(Y-X鈮�L3)

The region within (0,L/2)脳(L/2,L)where is satisfied localid="1647352406431" y-x鈮�L3is a right angle triangle with base x鈭�(L/6,L/2)andy鈭�(L/2,5L/6). Hence the area of that triangle is 12.

So,

P(Y-X鈮�L3)=4L2.12.(L3)2=29

we getP(Y-X鈮�L3)=29.

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