Question

Suppose that the joint probability density function of the random variables X and Y is f(x, y) = { 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 Var[XY]

          Suppose that the joint probability density function of the random variables X and Y is
f(x, y) = { 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 Var[XY]

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Suppose that the joint probability density function of the random variables X and Y is
f(x, y) = 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 Var[XY]

Added by Juan Jos- S.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Lucas Finney

Step 1

First, we need to find the value of c. To do this, we need to use the fact that the total probability should be equal to 1. So, we need to integrate the joint probability density function over the given region and set it equal to 1: $$\int_0^1 \int_0^1 (2 + cy^2) Show more…

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Suppose that the joint probability density function of the random variables X and Y is f(x,y) = { 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 Var[XY]

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