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4. Let S be the shadowed region as in the figure below: (c) Calculate E(Y |...
sucCesses isk. 6142 Let S consist of the 10 decimal digits. Suppose that a number X...
sucCesses isk. 6142 Let S consist of the 10 decimal digits. Suppose that a number X is chosen according to the discrete uniform distribution on S and then a number Y is chosen according to the discrete uniform distribution on {X, X +1,.. . .9}. a) Determine the conditional PMF of X given Y marginal PMF of Y and the joint PDF of X and Y b) Evaluate the conditional PMF in part (a), given Y = 6. = y...
1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a....
1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a. Find the joint pdf fxy(x, y), marginal pdf (fx(x) and fy()) and cdf (Fx(x) and Fy)) b. Find the conditional pdf fxiy Cr ly c. Find the probability P(X < Y = y) d. Are X and Y independent?
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let...
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
Let the continuous random variables X and (0, 2) and (3, 0). Y have a joint...
Let the continuous random variables X and (0, 2) and (3, 0). Y have a joint PDF which is uniform over the trig (U,0 a. Find the joint PDF of X and Y b. Find the marginal PDF of Y c. Find the conditional PDF of Xgiven Y. d. Find EIY/X x]
0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then fi...
0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then find the constant k and sketch f(r.y) in three dimensions (b) Find and sketch the marginal pdf fx), the conditional pdf(x1/2) and the conditional cdf FO11/2). Find P(X〈Y! Y〉 1/2), E(XİY=1/2) and E(XIY〉l/2). (c) What is the correlation between X...
Let X and Y be continuous random variables with joint pdf f(x,y) =fX (c(X + Y),...
Let X and Y be continuous random variables with joint pdf f(x,y) =fX (c(X + Y), 0 < y < x <1 otBerwise a. Find c. b. Find the joint pdf of S = Y and T = XY. c. Find the marginal pdf of T. 、
Suppose (X,Y ) is chosen according to the continuous uniform distribution on the triangle with vertices (0,0), (0,1) and (2,0), that is, the joint pdf of (X,Y ) is fX,Y (x,y) =c, for 0 ≤ x ≤ 2,0 ≤ y ≤...
Suppose (X,Y ) is chosen according to the continuous uniform distribution on the triangle with vertices (0,0), (0,1) and (2,0), that is, the joint pdf of (X,Y ) is fX,Y (x,y) =c, for 0 ≤ x ≤ 2,0 ≤ y ≤ 1, 1/ 2x + y ≤ 1, 0 , else. (a) Find the value of c. (b) Calculate the pdf, the mean and variance of X. (c) Calculate the pdf and the mean of Y . (d) Calculate the...
MA2500/18 Section B (Answer THREE questions) 6. Let X and Y be jointly continuous random variables...
MA2500/18 Section B (Answer THREE questions) 6. Let X and Y be jointly continuous random variables defined on the same prob- ability space, let fx.y denote their joint PDF, and let fx and fy respectively denote their marginal PDFs (a) Let z be a fixed value such that fx(x) >0. Write down expressions for 12] (i) the conditional PDF of Y given X = z, and (i) the conditional expectation of Y given X (b) State and prove the law...
please show all steps. Problem 23. Let the random variables X and Y have a joint...
please show all steps.
Problem 23. Let the random variables X and Y have a joint PDF which is uniform over the triangle with vertices at (0,0), (0,1), and (1.0). (a) Find the joint PDF of X and Y. (b) Find the marginal PDF of Y. (c) Find the conditional PDF of X given Y. (d) Find E[X|Y = y), and use the total expectation theorem to find E[X] in terms of E(Y). (e) Use the symmetry of the problem...
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...
sucCesses isk. 6142 Let S consist of the 10 decimal digits. Suppose that a number X is chosen according to the discrete uniform distribution on S and then a number Y is chosen according to the discrete uniform distribution on {X, X +1,.. . .9}. a) Determine the conditional PMF of X given Y marginal PMF of Y and the joint PDF of X and Y b) Evaluate the conditional PMF in part (a), given Y = 6. = y...
1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a. Find the joint pdf fxy(x, y), marginal pdf (fx(x) and fy()) and cdf (Fx(x) and Fy)) b. Find the conditional pdf fxiy Cr ly c. Find the probability P(X < Y = y) d. Are X and Y independent?
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
Let the continuous random variables X and (0, 2) and (3, 0). Y have a joint PDF which is uniform over the trig (U,0 a. Find the joint PDF of X and Y b. Find the marginal PDF of Y c. Find the conditional PDF of Xgiven Y. d. Find EIY/X x]
0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then find the constant k and sketch f(r.y) in three dimensions (b) Find and sketch the marginal pdf fx), the conditional pdf(x1/2) and the conditional cdf FO11/2). Find P(X〈Y! Y〉 1/2), E(XİY=1/2) and E(XIY〉l/2). (c) What is the correlation between X...
Let X and Y be continuous random variables with joint pdf f(x,y) =fX (c(X + Y), 0 < y < x <1 otBerwise a. Find c. b. Find the joint pdf of S = Y and T = XY. c. Find the marginal pdf of T. 、
MA2500/18 Section B (Answer THREE questions) 6. Let X and Y be jointly continuous random variables defined on the same prob- ability space, let fx.y denote their joint PDF, and let fx and fy respectively denote their marginal PDFs (a) Let z be a fixed value such that fx(x) >0. Write down expressions for 12] (i) the conditional PDF of Y given X = z, and (i) the conditional expectation of Y given X (b) State and prove the law...
please show all steps.
Problem 23. Let the random variables X and Y have a joint PDF which is uniform over the triangle with vertices at (0,0), (0,1), and (1.0). (a) Find the joint PDF of X and Y. (b) Find the marginal PDF of Y. (c) Find the conditional PDF of X given Y. (d) Find E[X|Y = y), and use the total expectation theorem to find E[X] in terms of E(Y). (e) Use the symmetry of the problem...
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...
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