Let the random variables X and Y have a joint PDF which is uniform over the triangle with verticies at (0,0),(0,1), and (1,0).Find the joint PDF of X and YFind the marginal PDF of YFInd the condtional PDF of X given YFind E[X[Y=y], and use the total expectation theorem to find E[X] in terms of E[Y]Use the symmetry of the problem to find the value of E[X]

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Asked Oct 15, 2019
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Let the random variables X and Y have a joint PDF which is uniform over the triangle with verticies at (0,0),(0,1), and (1,0).

Find the joint PDF of X and Y

Find the marginal PDF of Y

FInd the condtional PDF of X given Y

Find E[X[Y=y], and use the total expectation theorem to find E[X] in terms of E[Y]

Use the symmetry of the problem to find the value of E[X]

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Expert Answer

Step 1

Hey, since there are multiple subpart questions posted, we will answer first three subparts question. If you want any specific subpart question to be answered then please submit that question only or specify the question number in your message.

Step 2

From the given information, random variables X and Y follows uniform distribution. The joint probability density function is uniform over the triangle with verticies at (0,0),(0,1), and (1,0).

The triangle is,

D
(0, 1)
у
(0, 00
(1, 0)
х
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Image Transcriptionclose

D (0, 1) у (0, 00 (1, 0) х

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Step 3

The joint pdf of X and Y is obtained below:

Here the distribution is uniformly distributed,  f(x,y)=k, were,  K€R.

The area of a triangle formula is (1/2)(base)(height).

The area of ∆OCD is (1/2)(x)(y)=(1/2)(1)(1)=(1/2)

The limits of x and y are,

 (1/2)xy≤(1/2)

So xy≤1 and 0≤xy≤1—(1)

According to triangle principle

√z(x^2+y^2)≤1

0≤(x^2+y^2)≤1---(2)

Adding (1) and (2) we get,

0≤(x+y)^2≤2

0≤x+y≤ √2

Therefore, the limits of x and y are

0≤x≤√2 and 0≤y≤√2

The required joi...

(xy) dxcdy=
kdxdy =1
x y
kxx11
k 2
The Joint pdf of Xand Yis,
2 ,0sx0sys
(%) 0, therwise
help_outline

Image Transcriptionclose

(xy) dxcdy= kdxdy =1 x y kxx11 k 2 The Joint pdf of Xand Yis, 2 ,0sx0sys (%) 0, therwise

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