Question

# Suppose that the joint probability density function of the random variables X and Y is f(x,...

Suppose that the joint probability density function of the random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.

(a) Sketch the region of non-zero probability density and show that c = 3/ 2 .

(b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1).

(c) Compute the marginal density function of X and Y and hence calculate E [X] and E [Y ].

(d) Find the conditional density function of X given Y = y.

(e) Determine the covariance between X and Y , cov [X, Y ]=E[XY ]E [X]E[Y ].

(f) State, giving reasons, whether X and Y are independent. (g) Find V ar[XY ]

The Joint density function of the random variables X and Y is a) The region is shown below Now as total probability is 1, so b) As , X and Y are continuous random variables so their sum is also continuous .

Hence,  c) The marginal PDF's are given by and Hence, and d) Using definition of conditional PDF we get NOTE: As per rules, only 4 parts will be solved at a time

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