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
As , X and Y are continuous random variables so their sum is also continuous .
c) The marginal PDF's are given by
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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