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3. Let (X. X2) be standard bivariate normal with p = 3/5. Let (Y.Y2) be the...
3. Let X ~ N2(u, ) be a bivariate Normal random vector, where 6-61-10 X =...
3. Let X ~ N2(u, ) be a bivariate Normal random vector, where 6-61-10 X = = | 12 (a) Find the distribution of Y1 = X1 + X2. (b) Let Y2 = X1 +aX2. Find value a such that Yį and Y2 are independent.
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint pr...
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12...
2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12 X2 = 321 - 22 +11. (a) Find the joint density function of (X1, X2). (b) Find the covariance of X1 and X2. Now let Y1 = X1 + 4X2 +3 Y, = -2X2 +6X2 +5 (a) Find the joint density function of (Y1, Y). (b) Find the covariance of Yi and Y2.
Let X = (X1, X2) be a bivariate normal random vector such that Mi = 4,42...
Let X = (X1, X2) be a bivariate normal random vector such that Mi = 4,42 = 6,01 = 25, 02 = 16 and p= 0.7. 1. Find P(X2 <5|X1 = 3).
= = 3, Cov(X1, X2) = 2, Cov(X2, X3) = -2, Let Var(X1) = Var(X3) =...
= = 3, Cov(X1, X2) = 2, Cov(X2, X3) = -2, Let Var(X1) = Var(X3) = 2, Var(X2) Cov(X1, X3) = -1. i) Suppose Y1 = X1 - X2. Find Var(Y1). ii) Suppose Y2 = X1 – 2X2 – X3. Find Var(Y2) and Cov(Yı, Y2). Assuming that (X1, X2, X3) are multivariate normal, with mean 0 and covariances as specified above, find the joint density function fxı,Y,(y1, y2). iii) Suppose Y3 = X1 + X2 + X3. Compute the covariance...
Example 3 The scores on a midterm exam follow a normal distribution with an average of...
Example 3 The scores on a midterm exam follow a normal distribution with an average of 80.4% and a standard deviation of 10.9%. Let X represent the score of a given student on this midterm exam. 1. What is the probability that a randomly selected student scores above a 90%? 2. What is the probability that a randomly selected student scores between 80% and 90%? 3. What is the 60th percentile score for this midterm exam?
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be...
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
(2) Given two independent variables X1 and X2 having Bernoulli distribution with parameter p=1/3, let Y1...
(2) Given two independent variables X1 and X2 having Bernoulli distribution with parameter p=1/3, let Y1 = 2X1 and Y2 = 2X2. Then A E[Y1 · Y2] = 2/9 BE[Y1 · Y2] = 4/9 C P[Y1 · Y2 = 0) = 1/9 D P[Y1 · Y2 = 0) = 2/9 (3) Let X and Y be two independent random variables having gaussian (normal) distribution with mean 0 and variance equal 2. Then: A P[X +Y > 2] > 0.5 B...
MULTIVARIATE DISTRIBUTIONS 3. Suppose that Xi and X2 are independent and each has a uniform distribution...
MULTIVARIATE DISTRIBUTIONS
3. Suppose that Xi and X2 are independent and each has a uniform distribution on (0,1). Define Y: X1 + X2 and Y2 = X1-X2. Find the marginal probability density functions of Y1 and Y2. . Suppose that X has a standard normal distribution, and that the conditional distribution of Y given X is a normal distribution with mean 2X 3 and variance 12. Find E(Y) and Var(Y)
Graybill, 1961]. Let x -(X1, X2) have a bivariate normal distribution with pdf where Q-2x-x1x2 +...
Graybill, 1961]. Let x -(X1, X2) have a bivariate normal distribution with pdf where Q-2x-x1x2 + 4 _ 11x1-5T2 + 19, and k is a constant. Find a constant a such that P(3X1-X2 < a) 0.01.
3. Let X ~ N2(u, ) be a bivariate Normal random vector, where 6-61-10 X = = | 12 (a) Find the distribution of Y1 = X1 + X2. (b) Let Y2 = X1 +aX2. Find value a such that Yį and Y2 are independent.
2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12 X2 = 321 - 22 +11. (a) Find the joint density function of (X1, X2). (b) Find the covariance of X1 and X2. Now let Y1 = X1 + 4X2 +3 Y, = -2X2 +6X2 +5 (a) Find the joint density function of (Y1, Y). (b) Find the covariance of Yi and Y2.
Let X = (X1, X2) be a bivariate normal random vector such that Mi = 4,42 = 6,01 = 25, 02 = 16 and p= 0.7. 1. Find P(X2 <5|X1 = 3).
= = 3, Cov(X1, X2) = 2, Cov(X2, X3) = -2, Let Var(X1) = Var(X3) = 2, Var(X2) Cov(X1, X3) = -1. i) Suppose Y1 = X1 - X2. Find Var(Y1). ii) Suppose Y2 = X1 – 2X2 – X3. Find Var(Y2) and Cov(Yı, Y2). Assuming that (X1, X2, X3) are multivariate normal, with mean 0 and covariances as specified above, find the joint density function fxı,Y,(y1, y2). iii) Suppose Y3 = X1 + X2 + X3. Compute the covariance...
Example 3 The scores on a midterm exam follow a normal distribution with an average of 80.4% and a standard deviation of 10.9%. Let X represent the score of a given student on this midterm exam. 1. What is the probability that a randomly selected student scores above a 90%? 2. What is the probability that a randomly selected student scores between 80% and 90%? 3. What is the 60th percentile score for this midterm exam?
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
(2) Given two independent variables X1 and X2 having Bernoulli distribution with parameter p=1/3, let Y1 = 2X1 and Y2 = 2X2. Then A E[Y1 · Y2] = 2/9 BE[Y1 · Y2] = 4/9 C P[Y1 · Y2 = 0) = 1/9 D P[Y1 · Y2 = 0) = 2/9 (3) Let X and Y be two independent random variables having gaussian (normal) distribution with mean 0 and variance equal 2. Then: A P[X +Y > 2] > 0.5 B...
MULTIVARIATE DISTRIBUTIONS
3. Suppose that Xi and X2 are independent and each has a uniform distribution on (0,1). Define Y: X1 + X2 and Y2 = X1-X2. Find the marginal probability density functions of Y1 and Y2. . Suppose that X has a standard normal distribution, and that the conditional distribution of Y given X is a normal distribution with mean 2X 3 and variance 12. Find E(Y) and Var(Y)
Graybill, 1961]. Let x -(X1, X2) have a bivariate normal distribution with pdf where Q-2x-x1x2 + 4 _ 11x1-5T2 + 19, and k is a constant. Find a constant a such that P(3X1-X2 < a) 0.01.
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