# 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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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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Step 1

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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,

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...

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