4. Let (, F, P) be a probability space and let X and Y be random variables on it. Assume that the random variable X takes on value 0 and 1, and the distribution of X is given by Px (0)=0.3, Px(1) = 0.7. Further assume that the random variable Y takes on value 0 and 1, and the distribution of Y is given by Py (0)=0.8, Py(1) = 0.2. (i) Assume that the random variables X and Y are independent. Find the joint distribution of X and Y, Px,y. [5 Marks] (ii) Give an example of a possible joint distribution Px,y such that the random variables X, Y are not independent. [15 Marks] Hint. Denote Px,y (0, 0) = = a, Px,y (0, 1) = b, Px,y(1, 0) = = C, Px,y (1, 1) = = d. We must have (why?) a, b, c, d = [0, 1], a+b+c+d= 1, a+b= 0.3, a+c=0.8. Choose a, b, c, d satisfying the above conditions and such that a 0.24.
4. Let (, F, P) be a probability space and let X and Y be random variables on it. Assume that the random variable X takes on value 0 and 1, and the distribution of X is given by Px (0)=0.3, Px(1) = 0.7. Further assume that the random variable Y takes on value 0 and 1, and the distribution of Y is given by Py (0)=0.8, Py(1) = 0.2. (i) Assume that the random variables X and Y are independent. Find the joint distribution of X and Y, Px,y. [5 Marks] (ii) Give an example of a possible joint distribution Px,y such that the random variables X, Y are not independent. [15 Marks] Hint. Denote Px,y (0, 0) = = a, Px,y (0, 1) = b, Px,y(1, 0) = = C, Px,y (1, 1) = = d. We must have (why?) a, b, c, d = [0, 1], a+b+c+d= 1, a+b= 0.3, a+c=0.8. Choose a, b, c, d satisfying the above conditions and such that a 0.24.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.7: Probability
Problem 39E: Assume that the probability that an airplane engine will fail during a torture test is 12and that...
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![4. Let (, F, P) be a probability space and let X and Y be random variables
on it. Assume that the random variable X takes on value 0 and 1, and the
distribution of X is given by
Px (0)=0.3, Px(1) = 0.7.
Further assume that the random variable Y takes on value 0 and 1, and the
distribution of Y is given by
Py (0)=0.8, Py(1) = 0.2.
(i) Assume that the random variables X and Y are independent. Find
the joint distribution of X and Y, Px,y.
[5 Marks]
(ii) Give an example of a possible joint distribution Px,y such that the
random variables X, Y are not independent.
[15 Marks]
Hint. Denote
Px,y (0, 0) =
= a,
Px,y (0, 1) = b, Px,y(1, 0) =
= C,
Px,y (1, 1) =
= d.
We must have (why?)
a, b, c, d = [0, 1],
a+b+c+d= 1,
a+b= 0.3,
a+c=0.8.
Choose a, b, c, d satisfying the above conditions and such that a 0.24.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39a0c022-e49a-477b-ab1a-da6fafebfbe8%2F23a4a1ea-bb2d-4957-a99c-afe741e79c8c%2F2q8x75d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Let (, F, P) be a probability space and let X and Y be random variables
on it. Assume that the random variable X takes on value 0 and 1, and the
distribution of X is given by
Px (0)=0.3, Px(1) = 0.7.
Further assume that the random variable Y takes on value 0 and 1, and the
distribution of Y is given by
Py (0)=0.8, Py(1) = 0.2.
(i) Assume that the random variables X and Y are independent. Find
the joint distribution of X and Y, Px,y.
[5 Marks]
(ii) Give an example of a possible joint distribution Px,y such that the
random variables X, Y are not independent.
[15 Marks]
Hint. Denote
Px,y (0, 0) =
= a,
Px,y (0, 1) = b, Px,y(1, 0) =
= C,
Px,y (1, 1) =
= d.
We must have (why?)
a, b, c, d = [0, 1],
a+b+c+d= 1,
a+b= 0.3,
a+c=0.8.
Choose a, b, c, d satisfying the above conditions and such that a 0.24.
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