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Problem 1: Let X and Y have joint density function f(x, y) = { x+y for 0 < x, y < 1; 0 otherwise } a. Verify that f is indeed a joint probability density function. b. Find the marginal probability function f_X(x) of the random variable X. Use your result to compute the marginal mean ?_X and variance ?_X^2 of X. c. Find the marginal probability function f_Y(Y) of the random variable Y. Use your result to compute the marginal mean ?_Y and variance ?_Y^2 of Y. d. Compute the product moment E[X, Y] of X and Y. e. Compute the covariance and the correlation coefficient between X and Y. f. Find the conditional probability density of Y, given the event X = x, h(y/x), and compute the conditional expectation E[Y/X = 1/2].

          Problem 1:
Let X and Y have joint density function
f(x, y) = { x+y for 0 < x, y < 1; 0 otherwise }
a. Verify that f is indeed a joint probability density function.
b. Find the marginal probability function f_X(x) of the random variable X. Use your result to compute the marginal mean ?_X and variance ?_X^2 of X.
c. Find the marginal probability function f_Y(Y) of the random variable Y. Use your result to compute the marginal mean ?_Y and variance ?_Y^2 of Y.
d. Compute the product moment E[X, Y] of X and Y.
e. Compute the covariance and the correlation coefficient between X and Y.
f. Find the conditional probability density of Y, given the event X = x, h(y/x), and compute the conditional expectation E[Y/X = 1/2].

Problem 1:
Let X and Y have joint density function
f(x, y) = x+y for 0 < x, y < 1; 0 otherwise
a. Verify that f is indeed a joint probability density function.
b. Find the marginal probability function fX(x) of the random variable X. Use your result to compute the marginal mean ?X and variance ?X^2 of X.
c. Find the marginal probability function fY(Y) of the random variable Y. Use your result to compute the marginal mean ?Y and variance ?Y^2 of Y.
d. Compute the product moment E[X, Y] of X and Y.
e. Compute the covariance and the correlation coefficient between X and Y.
f. Find the conditional probability density of Y, given the event X = x, h(y/x), and compute the conditional expectation E[Y/X = 1/2].

Added by Sarah O.

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