Suppose that X is uniformly distributed between 0 and 1. Given X = x, Y is...
Suppose that X is uniformly distributed between 0 and 1. Given X = x, Y is uniformly distributed between 0 and x2.
(a) Determine E(Y |X = x) and then Var(Y |X = x). Is E(Y |X = x) a linear function of x?
(b) Find f(x, y) using fX(x) and fY |X(y|x).
(c) Find fY (y).
(d) Find the conditional density of X given Y = y.
(e) Find the correlation coefficient between X and Y .
Homework Answers
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Suppose that X is uniformly distributed between 0 and 1. Given X
= x, Y is...
Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given...
Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given that Y -y is uniform on (0, y). Find E[X] and Var(X).
Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given...
Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given that Y -y is uniform on (0, y). Find E[X] and Var(X).
Suppose Y is uniformly distributed on (0,1), and that the conditional distribution of X given that...
Suppose Y is uniformly distributed on (0,1), and that the conditional distribution of X given that Y = y is uniform on (0, y). Find E[X]and Var(X).
Need help with question 2 (not question 1) 1. Suppose that (X,Y) is uniformly distributed over...
Need help with question 2 (not question
1)
1. Suppose that (X,Y) is uniformly distributed over the region {(x, y): 0 < \y< x < 1}. Find: a) the joint density of (X, Y); b) the marginal densities fx(x) and fy(y). c) Are X and Y independent? d) Find E(X) and E(Y). 2. Repeat Exercise 1 for (X,Y) with uniform distribution over {(x, y): 0 < \x]+\y< 1}.
Exercise 10.33. Let (X,Y) be uniformly distributed on the triangleD with vertices (1,0), (2,0) and (0,1), as in Example...
Exercise 10.33. Let (X,Y) be uniformly distributed on the
triangleD with vertices (1,0), (2,0) and (0,1), as in Example
10.19. (a) Find the conditional probability P(X ≤ 1 2|Y =y). You
might first deduce the answer from Figure 10.2 and then check your
intuition with calculation. (b) Verify the averaging identity for
P(X ≤ 1 2). That is, check that P(X ≤ 1 2)=:∞ −∞ P(X ≤ 1 2|Y
=y)fY(y)dy.
Example 10.19. Let (X, Y) be uniformly distributed on the...
The joint probability density function (pdf) of (X,Y ) is given by f(X,Y )(x,y) = 12/ 7 x(x + y), for 0 ≤ y ≤ 1, 0 ≤ x ≤ 1, 0, elsewhere. (a) Find the cumulative distribution function of (X,Y ). Make...
The joint probability density function (pdf) of (X,Y ) is given by f(X,Y )(x,y) = 12/ 7 x(x + y), for 0 ≤ y ≤ 1, 0 ≤ x ≤ 1, 0, elsewhere. (a) Find the cumulative distribution function of (X,Y ). Make sure you derive expressions for the cdf in the regions • x < 0 or y < 0; • 0 ≤ x ≤ 1, 0 ≤ y ≤ 1; • x > 1, 0 ≤ y ≤...
Show the random variables X and Y are independent, or not independent Find the joint cdf given the joint pdf below Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 a...
Show the random variables X and Y are independent, or not
independent
Find the joint cdf given the joint pdf below
Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 and -1sx 4 Therefore, the joint probability density function is, 0; Otherwise
Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 and -1sx 4
Therefore, the joint probability density function is, 0; Otherwise
4. Suppose X and Y have the joint pdf f(x,y) = 6x, 0 < x <...
4. Suppose X and Y have the joint pdf f(x,y) = 6x, 0 < x < y < 1, and zero otherwise. (a) Find fx(x). (b) Find fy(y). (c) Find Corr(X,Y). (d) Find fy x(y|x). (e) Find E(Y|X). (f) Find Var(Y). (g) Find Var(E(Y|X)). (h) Find E (Var(Y|X)]. (i) Find the pdf of Y - X.
Problem 5. Suppose that a uniformly distributed random number X in 0 is found by calling...
Problem 5. Suppose that a uniformly distributed random number X in 0 is found by calling a random number generator. Then, if the call to the RNG pro- duces the value r for X, another random umber Y is computed that is uniformly distributed on 0, . That is, X is uniform on the interval 0,1], and the conditional distribution for Y given X = 1 is uniform on the interval [0.11 a) Give fonmulas for E(Y X) and Var(Y...
4. The random variables X and Y have joint probability density function fx,y(x, ) given by: fx,y(...
4. The random variables X and Y have joint probability density function fx,y(x, ) given by: fx,y(x, y) 0, else (a) Find c. (b) Find fx(x) and fy (), the marginal probability density functions of X and Y, respectively (c) Find fxjy (xly), the conditional probability density function of X given Y. For your limits (which you should not forget!), put y between constant bounds and then give the limits for in terms of y. (d) Are X and Y...
Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given that Y -y is uniform on (0, y). Find E[X] and Var(X).
Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given that Y -y is uniform on (0, y). Find E[X] and Var(X).
Need help with question 2 (not question
1)
1. Suppose that (X,Y) is uniformly distributed over the region {(x, y): 0 < \y< x < 1}. Find: a) the joint density of (X, Y); b) the marginal densities fx(x) and fy(y). c) Are X and Y independent? d) Find E(X) and E(Y). 2. Repeat Exercise 1 for (X,Y) with uniform distribution over {(x, y): 0 < \x]+\y< 1}.
Exercise 10.33. Let (X,Y) be uniformly distributed on the
triangleD with vertices (1,0), (2,0) and (0,1), as in Example
10.19. (a) Find the conditional probability P(X ≤ 1 2|Y =y). You
might first deduce the answer from Figure 10.2 and then check your
intuition with calculation. (b) Verify the averaging identity for
P(X ≤ 1 2). That is, check that P(X ≤ 1 2)=:∞ −∞ P(X ≤ 1 2|Y
=y)fY(y)dy.
Example 10.19. Let (X, Y) be uniformly distributed on the...
Show the random variables X and Y are independent, or not
independent
Find the joint cdf given the joint pdf below
Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 and -1sx 4 Therefore, the joint probability density function is, 0; Otherwise
Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 and -1sx 4
Therefore, the joint probability density function is, 0; Otherwise
4. Suppose X and Y have the joint pdf f(x,y) = 6x, 0 < x < y < 1, and zero otherwise. (a) Find fx(x). (b) Find fy(y). (c) Find Corr(X,Y). (d) Find fy x(y|x). (e) Find E(Y|X). (f) Find Var(Y). (g) Find Var(E(Y|X)). (h) Find E (Var(Y|X)]. (i) Find the pdf of Y - X.
Problem 5. Suppose that a uniformly distributed random number X in 0 is found by calling a random number generator. Then, if the call to the RNG pro- duces the value r for X, another random umber Y is computed that is uniformly distributed on 0, . That is, X is uniform on the interval 0,1], and the conditional distribution for Y given X = 1 is uniform on the interval [0.11 a) Give fonmulas for E(Y X) and Var(Y...
4. The random variables X and Y have joint probability density function fx,y(x, ) given by: fx,y(x, y) 0, else (a) Find c. (b) Find fx(x) and fy (), the marginal probability density functions of X and Y, respectively (c) Find fxjy (xly), the conditional probability density function of X given Y. For your limits (which you should not forget!), put y between constant bounds and then give the limits for in terms of y. (d) Are X and Y...
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