Question

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]

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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