Homework Answers
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Here we can't find the value
of a because Y1 and Y2 are not independent.Since
X1 and X2 are themeselves not independent
therefore its linear combination is also not independent
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3. Let X ~ N2(u, ) be a bivariate Normal random vector, where 6-61-10 X =...
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5...
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5 ], = [11] Here -1 <p<1. (a) What is P(X > Y)? (b) Is there a constant c such that X and X +cY are independent?
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).
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 (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) .
3. Let (X. X2) be standard bivariate normal with p = 3/5. Let (Y.Y2) be the...
3. Let (X. X2) be standard bivariate normal with p = 3/5. Let (Y.Y2) be the midterm and final exam scores of a randomly selected student. Assume Y1 = 80 +3X1Y2 = 75 + 2X2. Given a student got 90 in the midterm exam, (a) What is the conditional expectation and conditional variance of her final exam score? Hint. Probably easier to reduce the question to (X1, X2) but also (Y1. Y2) is a normal bivariate. (b) What is the...
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y;...
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
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 ....
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.
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 -...
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent.
Unif (0, 1) 5. Suppose U1 and...
The random variables Y1 and Y2 follow the bivariate normal distribution in (2.74). Show that if...
The random variables Y1 and Y2 follow the
bivariate normal distribution in (2.74). Show that if 12 =
0, Y1, and Y2 are independent random
variables.
We were unable to transcribe this imageexp1 21 Pí2 (2.74) 2p12 σ2 σ2
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5 ], = [11] Here -1 <p<1. (a) What is P(X > Y)? (b) Is there a constant c such that X and X +cY are independent?
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).
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.
3. Let (X. X2) be standard bivariate normal with p = 3/5. Let (Y.Y2) be the midterm and final exam scores of a randomly selected student. Assume Y1 = 80 +3X1Y2 = 75 + 2X2. Given a student got 90 in the midterm exam, (a) What is the conditional expectation and conditional variance of her final exam score? Hint. Probably easier to reduce the question to (X1, X2) but also (Y1. Y2) is a normal bivariate. (b) What is the...
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
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 ....
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.
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent.
Unif (0, 1) 5. Suppose U1 and...
The random variables Y1 and Y2 follow the
bivariate normal distribution in (2.74). Show that if 12 =
0, Y1, and Y2 are independent random
variables.
We were unable to transcribe this imageexp1 21 Pí2 (2.74) 2p12 σ2 σ2
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