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.


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


The joint pdf of X is

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

= 0 otherwise.


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


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.


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
A cantilever beam is loaded with a concentrated. Force of 5kN at the free end. The length of the cantilever beam is 1m having cross section of 50×50mm. determine the equation for b

Define Radiography - Underwater Inspection of Bridge? Radiographic non-destructive examination of components is based on the phenomenon that materials absorb radiation energy a

Question How does an engineer determine digit of cells for concrete box girder bridges ? Answer If depth of a box girder bridge exceeds 1/6 or 1/5 of bridge width, then it is o

A group of 16 piles of 50 cm diameter is arranged with a centre to centre spacing of 1.0 m. The piles are 9 m long and are embedded in soft clay with cohesion 30 kN/m 2 . Bearing r

Compass Surveying: You have studied about the compass and its use for bearings of measuring survey lines. The angular measurements are carried out with prismatic compass that

what is importance of positioning in different types of engineering?

Question What is major disparity between fasteners , bolts and screws ? Answer Fastener is a broad term to explain something that is used as a restraint for holding

Design a sewer network

Calculate the Amount of Waste A municipal solid waste department plans to separate a portion of the ferrous metal, newsprint and cardboard from its MSW waste stream. The depart

Q. Case of Multi speed restrictions? Position of Engineering indicators in case of Multi speed restrictions on one of the line in a Double line section where the first speed re