3. Let X and Y be random variables on a probability space (Q, F, P) that take on values in [0,3] and [0,2], respectively. Assume that the joint distribu- tion of X and Y on [0,3] × [0,2] is given by 1 dPx,y(x, y) = (x² + y³) dx dy. 30 (i) Calculate the joint moments of the random variables X and Y, i.e., E(XY) for m, n = No = {0, 1, 2,...}. (ii) Find the covariance between X and Y, Cov(X, Y). [20 Marks] [5 Marks] (iii) Find the distribution of the random variable X and the distribution of the random variable Y. [10 Marks] (iv) Are the random variables X and Y independent? Justify your answer. (v) Find the probability P(3Y ≤ 2X). [5 Marks] [20 Marks]
3. Let X and Y be random variables on a probability space (Q, F, P) that take on values in [0,3] and [0,2], respectively. Assume that the joint distribu- tion of X and Y on [0,3] × [0,2] is given by 1 dPx,y(x, y) = (x² + y³) dx dy. 30 (i) Calculate the joint moments of the random variables X and Y, i.e., E(XY) for m, n = No = {0, 1, 2,...}. (ii) Find the covariance between X and Y, Cov(X, Y). [20 Marks] [5 Marks] (iii) Find the distribution of the random variable X and the distribution of the random variable Y. [10 Marks] (iv) Are the random variables X and Y independent? Justify your answer. (v) Find the probability P(3Y ≤ 2X). [5 Marks] [20 Marks]
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.4: Hyperbolas
Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
Related questions
Question
![3.
Let X and Y be random variables on a probability space (Q, F, P) that
take on values in [0,3] and [0,2], respectively. Assume that the joint distribu-
tion of X and Y on [0,3] × [0,2] is given by
1
dPx,y(x, y)
=
(x² + y³) dx dy.
30
(i) Calculate the joint moments of the random variables X and Y, i.e.,
E(XY) for m, n = No
=
{0, 1, 2,...}.
(ii) Find the covariance between X and Y, Cov(X, Y).
[20 Marks]
[5 Marks]
(iii) Find the distribution of the random variable X and the distribution
of the random variable Y.
[10 Marks]
(iv) Are the random variables X and Y independent? Justify your answer.
(v) Find the probability P(3Y ≤ 2X).
[5 Marks]
[20 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b616b6f-1a99-4661-b03e-d5650e4084c7%2F36c42506-70ff-4bf7-a023-df75f99e6f39%2Fjjfcg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.
Let X and Y be random variables on a probability space (Q, F, P) that
take on values in [0,3] and [0,2], respectively. Assume that the joint distribu-
tion of X and Y on [0,3] × [0,2] is given by
1
dPx,y(x, y)
=
(x² + y³) dx dy.
30
(i) Calculate the joint moments of the random variables X and Y, i.e.,
E(XY) for m, n = No
=
{0, 1, 2,...}.
(ii) Find the covariance between X and Y, Cov(X, Y).
[20 Marks]
[5 Marks]
(iii) Find the distribution of the random variable X and the distribution
of the random variable Y.
[10 Marks]
(iv) Are the random variables X and Y independent? Justify your answer.
(v) Find the probability P(3Y ≤ 2X).
[5 Marks]
[20 Marks]
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