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Introduction to Probability, | 2nd Edition | ISBN: 9781886529236 | Authors: Dimitri P. Bertsekas John N. Tsitsiklis
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Let the random variables X and Y have a joint PDF which is

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QUESTION:

Let the random variables X and Y have a joint PDF which is uniform over the triangle with vertices at (0, 0), (0, 1), and (1. 0). (a) Find the joint PDF of X and Y. (b) Find the marginal PDF of Y. (c) Find the conditional PDF of X given Y. (d) Find E[X I Y = yj, and use the total expectation theorem to find E[X] in terms of E[Y] . (e) Use the symmetry of the problem to find the value of E[X].

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QUESTION:

Let the random variables X and Y have a joint PDF which is uniform over the triangle with vertices at (0, 0), (0, 1), and (1. 0). (a) Find the joint PDF of X and Y. (b) Find the marginal PDF of Y. (c) Find the conditional PDF of X given Y. (d) Find E[X I Y = yj, and use the total expectation theorem to find E[X] in terms of E[Y] . (e) Use the symmetry of the problem to find the value of E[X].

ANSWER:

Step 1 of 5

(a)

We interpret the word “triangle” to mean the “closed triangle,” that is, its boundary is also included. The area of the triangle is , so that , on the triangle indicated in the above figure, and zero everywhere else.

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