Find out the joint distribution, Civil Engineering

Assignment Help:

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


Related Discussions:- Find out the joint distribution

Difference between anionic emulsions and cationic emulsions, Q. Bitumen emu...

Q. Bitumen emulsions - difference between anionic emulsions and cationic emulsions? Bitumen emulsions comprises particles of bitumen dispersed in water by using emulsifying age

Strength of materials, distribution of shear stress in triangular section

distribution of shear stress in triangular section

Formation of pedestrian level winds around buildings, Q. Formation of pedes...

Q. Formation of pedestrian level winds around buildings? When a building blocks the wind blowing across it, part of the wind will escape over the top of the building. Some will

Write short note on surplus store, Q. Write short note on Surplus Store? ...

Q. Write short note on Surplus Store? Comprise stores which are surplus to requirements and include materials released from renewals and replacements as well as from dismantled

Lime, LIME It is significant binding material utilized in building con...

LIME It is significant binding material utilized in building construction. Lime has been utilized as the material of construction from ancient time. While it is mixed with san

Strength of materials, what are the different between stress and strain

what are the different between stress and strain

Mr Kamal Saxena, What is WBM road. How it is constructed?

What is WBM road. How it is constructed?

Syphon, what is the design of syphon

what is the design of syphon

Wmm, what is material of wmm plant

what is material of wmm plant

Define pneumatic tools for underwater inspection of bridge, Define Pneumati...

Define Pneumatic Tools for Underwater Inspection of Bridge? These are more efficient than hand tools and cause less diver fatigue. The disadvantages are that below about 20 m,

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd