Find out the joint distribution, Civil Engineering

Find out the joint distribution:

Let XI and X2 be two independent random variables each distributed uniformly in the interval [ 0, a ], where a > 0 is a constant. Find out the joint distribution of

Yl = Xl + X2 and Y2 = X1 - X2.

Instead, in vector notation, what is the distribution of Y = XA.

where

x = (X1,X2),Y= (Y1,Y2), A = 584_Find out the joint distribution.png? Find also the marginal distributions of Y1 and Y2. ?

Solution:

The joint pdf of X is

fx(x) = 1/a2, (x1,x2)? R(x)

= 0 otherwise.

Where

R(x) = {(x1,x2):0 ≤ x1 ≤ a, 0 ≤  x2 ≤ a}

The Jacobian of the transformation is

347_Find out the joint distribution1.png

Hence the pdf of Y is

fy(y) = 1/2a2, (y1,y2)? R(y)

= 0 otherwise.

where R ( y ) is the transformed region R ( x ) under the transformation Y = XA. The range of variation of Yl is clearly [ 0,2a ] and that of Y2 is [ - a, + a ]. However Yl and Y2 are not independent.

Since the inverse transformation is

X1= ½ (Y1 + Y2), X2 = ½ (Y1 - Y2) and 0≤ x1, x2 ≤ a,

the region R ( y ) is given by

R(y) = {( Y1 + Y2) : 0 ≤ Y1 + Y2 ≤ 2a, 0≤ Y1 - Y2 ≤2a},

The Relation between R ( x ) and R ( y ) is illustrated in Figure 2.

1951_Find out the joint distribution2.png

Figure: Relation between R ( x ) and R ( y ).

Note that the variables xl and x2 are independent and the region R ( x ) is such that for Xl - xl, the variation X2 does not depend on xl, but the region R ( Y ) is not of that type and the transformed variables Yl and Y2 are not independent.

The variable Yl varies in the interval [ 0, 2a]and for a fixed yl, if 0≤ y1≤ a, then y2 takes on values -y1≤y2≤ y1, while, if a< y1≤ 2a then y2 varies in the interval

-(2a-y1) <.y2 ≤ (2a - y1)

Integrating fy ( y ) with respect to y2, the marginal pdf of y2 is obtained as follows

fY1(y1) = 2283_Find out the joint distribution3.png 1/2a2 dy2 = y1/a2, for 0 ≤ y1 ≤ a

462_Find out the joint distribution4.png 1/2a2 dy2  = 2a-y1/a2, for a< y1 ≤ 2a

= 0 otherwise.

In a similar manner, we note that for a given Y2, if -a ≤ y2 ≤ 0 then

-y2 ≤ y1 ≤ 2a-y2, and if 0≤ y2 ≤ a then y2 ≤ y1 ≤ 2a - y2

Hence,

fY2(y2) = 119_Find out the joint distribution5.png1/2a2 dy1 = a+y2/a2, -a ≤ y2 ≤ 0

960_Find out the joint distribution6.png 1/2a2 dy1 = a-y2/a2 , 0< y2 ≤ a

= 0 otherwise.

Remarks:

The forms of pdf the marginal distributions In Example 5 are shown in Figure 3. Due to their triangular shape of pdf's, the distributions are called triangular distributions.

2222_Find out the joint distribution7.png

 

Figure: The forms of the marginal distributions of YI and Y2

Posted Date: 1/30/2013 7:16:52 AM | Location : United States







Related Discussions:- Find out the joint distribution, Assignment Help, Ask Question on Find out the joint distribution, Get Answer, Expert's Help, Find out the joint distribution Discussions

Write discussion on Find out the joint distribution
Your posts are moderated
Related Questions
State the maximum amount of resistance in a pole The maximum amount of resistance in a pole is generally required at the base and, so, the maximum cross sectional area is requi

Q. Show the Functions of sleepers in railway? Functions of sleepers in railway works are as below:  (i)  The primary function of a sleeper is to grip the rail to gauge and t

Q. Illustrate the Design discharge for foundation? To provide for an adequate margin of safety, the foundation and protective works of a bridge should be designed for a flood d

A loading test was conducted with a 300 mm square plate at depth of 1 m under the ground surface in pure clay deposit. The water table is situated at a depth of 4 m below the groun

Question In determining effectual stress parameters of a soil sample, which test is good, consolidated undrained analysis or consolidated drained analysis? Answer Effec

Define Dual frequency and colour fathometer - Sounding/Sensing Devices? Dual Frequency and Colour Fathometers can be used to detect refill in the scoured area since more than o

can you help me with an assignment

Define the Cavitation - underwater inspection of bridge? Cavitation damage is caused by repeated impact forces caused by collapse of vapour bubbles in rapidly flowing water. Th