![Theorem 5.6. Given random variables X and Y, and constants a and b, VarſaX + bY] = aʼVar[X] + b2V ar[Y] + 2ab cov(X,Y)](http://img.homeworklib.com/questions/84dd5140-cb32-11ea-81a0-a3592b9e8723.png?x-oss-process=image/resize,w_560)
Homework Answers
![E(X) - 1.6 var(x) = E(X²) - [E(X)]2 - 2.80 - 1.62 = 2.80 - 2.56. var(x) = 0.25 YPLY) , y2P(4) 0.20 0.20 0-20 1 Y PLY) 10-20 2](http://img.homeworklib.com/questions/8705c580-cb32-11ea-90b7-e19987c70878.png?x-oss-process=image/resize,w_560)
So variance of W = X + Y is same for both the methods.
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Exercise 11. For the joint distribution given in Table 3, find the
variance of W =...
Suppose that the following table is the joint probability distribution of two random variables X and...
Suppose that the following table is the joint probability distribution of two random variables X and Y: х -2 0 2 3 0.27 0.08 0.16 0.2 0.1 0.04 0.1 0.05 a. Find the marginal PDF of X when x=-2, 0, 2, and 3. b. Find the marginal PDF of Y when y=2 and 5. . Find the conditional PDF of x=-2 and 3 given that y=2 has occurred. . Find the conditional PDF of y=2 and 5 given that x=3...
4. (20 points) Suppose the joint distribution of X and Y is: fxy(x, y) 1 0...
4. (20 points) Suppose the joint distribution of X and Y is: fxy(x, y) 1 0 1 2 3 0.04 0.06 0.01 0.00 0.13 0.13 0.02 0.12 0.04 0.06 0.00 0.11 0.07 0.10 0.06 (a) (4 points) Find the marginal distributions of X and Y. (b) (4 points) Given X = 3, what is the probability that random variable Y is at most 2?. (c) (4 points) Are random variables X and Y independent? Why or why not? (d) (4...
Use the probability distribution given in the table below and consider two new random variables, W=...
Use the probability distribution given in the table below and consider two new random variables, W= 1 + 9X and V = 4 + 2Y, to answer the following questions Joint Distribution of Weather Conditions and Commuting Times Long commute (Y = 0) Short commute (Y = 1) Total Rain (X = 0) 0.04 0.50 0.54 No Rain (X = 1) 0.32 0.14 0.46 Total 0.36 0.64 1.00 Compute the mean of W. E(W) = (Round your response to two...
6. (a) Find cov(W, Z) for W and Z defined in Problem 1. (b) The joint...
6. (a) Find cov(W, Z) for W and Z defined in Problem 1. (b) The joint density of random variables X and Y is f(x,y)-10,' elsewhere. Find cov(X, Y).
Given the following joint distribution of two random variables X and Y (a) Compute marginal distribution...
Given the following joint distribution of two random variables X
and Y
(a) Compute marginal distribution PX(x)
(b) Compute marginal distribution PY(y)
(c) What is the conditional probability P(Y | X = 2)?
20.10 0.05 0.15 0.10 0.10 4 0.04 0.02 0.06 0.04 0.04 6 0.04 0.02 0.06 0.06 0.02 8 0.02 0.01 0.03 0 0.04
Question 4: Let X and Y be two discrete random variables with the following joint probability...
Question 4: Let X and Y be two discrete random variables with the following joint probability distribution (mass) function Pxy(x, y): a) Complete the following probability table: Y 2 f(x)=P(X=x) 1 3 4 0 0 0.08 0.06 0.05 0.02 0.07 0.08 0.06 0.12 0.05 0.03 0.06 0.05 0.04 0.03 0.01 0.02 0.03 0.04 2 3 foy)=P(Y=y) 0.03 b) What is P(X s 2 and YS 3)? c) Find the marginal probability distribution (mass) function of X; [f(x)]. d) Find the...
The joint probability density function is f(x, y) for 17. Find the mean of X given...
The joint probability density function is f(x, y) for 17. Find the mean of X given Y = random variables X and Y fax, y) = f(xy *** Q<x<10<x<1 Elsewhere w 14. Random variables X and Y have a density function f(x, y). Find the indicated expected value f(x, y) = 6; (xy+y4) 0<x< 1,0<y<1 0 Elsewhere E(x2y) = 15. The means, standard deviations, and covariance for random variables X, Y, and Z are given below. Lex= 3, uy =...
3. Let X and Y have a discrete joint distribution with Table 1: Joint discrete distribution...
3. Let X and Y have a discrete joint distribution with Table 1: Joint discrete distribution of X and Y Values of Y -1 0 1 Values of X -1 1 į 0 1 1 0 -600-100 Then, find the following: • the marginal distribution of X; [2 points) • the marginal distribution of Y; [2 points] the conditional distribution of X given Y = -1; [2 points] Are X and Y are independent? Discuss with proper justification. (3 points)...
The discrete random variables X and Y take integer values with joint probability distribution given by...
The discrete random variables X and Y take integer values with joint probability distribution given by f (x,y) = a(y−x+1) 0 ≤ x ≤ y ≤ 2 or =0 otherwise, where a is a constant. 1 Tabulate the distribution and show that a = 0.1. 2 Find the marginal distributions of X and Y. 3 Calculate Cov(X,Y). 4 State, giving a reason, whether X and Y are independent. 5 Calculate E(Y|X = 1).
2) The random variables X and Y have the following joint distribution. 3 .25 2 .2...
2) The random variables X and Y have the following joint distribution. 3 .25 2 .2 1.16 1.04 .05 1.04 1.01 .05 a) Find the correlation between X and Y. b) Provide intuitive reasoning for your result from part a). 3) Take the joint distribution of X and Y from the previous exercise, and consider L, and L, to be lottery tickets that pay according to the following scheme: Li = X if X <2, L, = 5 otherwise. L=L+Y...
Suppose that the following table is the joint probability distribution of two random variables X and Y: х -2 0 2 3 0.27 0.08 0.16 0.2 0.1 0.04 0.1 0.05 a. Find the marginal PDF of X when x=-2, 0, 2, and 3. b. Find the marginal PDF of Y when y=2 and 5. . Find the conditional PDF of x=-2 and 3 given that y=2 has occurred. . Find the conditional PDF of y=2 and 5 given that x=3...
4. (20 points) Suppose the joint distribution of X and Y is: fxy(x, y) 1 0 1 2 3 0.04 0.06 0.01 0.00 0.13 0.13 0.02 0.12 0.04 0.06 0.00 0.11 0.07 0.10 0.06 (a) (4 points) Find the marginal distributions of X and Y. (b) (4 points) Given X = 3, what is the probability that random variable Y is at most 2?. (c) (4 points) Are random variables X and Y independent? Why or why not? (d) (4...
Use the probability distribution given in the table below and consider two new random variables, W= 1 + 9X and V = 4 + 2Y, to answer the following questions Joint Distribution of Weather Conditions and Commuting Times Long commute (Y = 0) Short commute (Y = 1) Total Rain (X = 0) 0.04 0.50 0.54 No Rain (X = 1) 0.32 0.14 0.46 Total 0.36 0.64 1.00 Compute the mean of W. E(W) = (Round your response to two...
6. (a) Find cov(W, Z) for W and Z defined in Problem 1. (b) The joint density of random variables X and Y is f(x,y)-10,' elsewhere. Find cov(X, Y).
Given the following joint distribution of two random variables X
and Y
(a) Compute marginal distribution PX(x)
(b) Compute marginal distribution PY(y)
(c) What is the conditional probability P(Y | X = 2)?
20.10 0.05 0.15 0.10 0.10 4 0.04 0.02 0.06 0.04 0.04 6 0.04 0.02 0.06 0.06 0.02 8 0.02 0.01 0.03 0 0.04
Question 4: Let X and Y be two discrete random variables with the following joint probability distribution (mass) function Pxy(x, y): a) Complete the following probability table: Y 2 f(x)=P(X=x) 1 3 4 0 0 0.08 0.06 0.05 0.02 0.07 0.08 0.06 0.12 0.05 0.03 0.06 0.05 0.04 0.03 0.01 0.02 0.03 0.04 2 3 foy)=P(Y=y) 0.03 b) What is P(X s 2 and YS 3)? c) Find the marginal probability distribution (mass) function of X; [f(x)]. d) Find the...
The joint probability density function is f(x, y) for 17. Find the mean of X given Y = random variables X and Y fax, y) = f(xy *** Q<x<10<x<1 Elsewhere w 14. Random variables X and Y have a density function f(x, y). Find the indicated expected value f(x, y) = 6; (xy+y4) 0<x< 1,0<y<1 0 Elsewhere E(x2y) = 15. The means, standard deviations, and covariance for random variables X, Y, and Z are given below. Lex= 3, uy =...
3. Let X and Y have a discrete joint distribution with Table 1: Joint discrete distribution of X and Y Values of Y -1 0 1 Values of X -1 1 į 0 1 1 0 -600-100 Then, find the following: • the marginal distribution of X; [2 points) • the marginal distribution of Y; [2 points] the conditional distribution of X given Y = -1; [2 points] Are X and Y are independent? Discuss with proper justification. (3 points)...
2) The random variables X and Y have the following joint distribution. 3 .25 2 .2 1.16 1.04 .05 1.04 1.01 .05 a) Find the correlation between X and Y. b) Provide intuitive reasoning for your result from part a). 3) Take the joint distribution of X and Y from the previous exercise, and consider L, and L, to be lottery tickets that pay according to the following scheme: Li = X if X <2, L, = 5 otherwise. L=L+Y...
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