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Let X be a standard normal distribution. Let ξ be another random variable, independent of X, whic...
xercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < 1...
xercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < 1 . Check that if X Z and Y-: ρΖ+ VI-P" W then the pair (X, Y) has standard bivariate normal distribution with parameter p. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.
Exercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < l....
Exercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < l. Check that if X-Z and Y-p2+ VI-p-W then the pair (X, Y) has standard bivariate normal distribution with parameter ρ. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.
Problem 1. Let X be a normal random variable with mean 0 and variance 1 and...
Problem 1. Let X be a normal random variable with mean 0 and variance 1 and let Y be uniform(0.1) with X and Y being independent. Let U-X + Y and V = X-Y. For this problem recall the density for a normal random variable is 2πσ2 (a) Find the joint distribution of U and V (b) Find the marginal distributions of U and V (c) Find Cov(U, V).
Suppose that X and Y are bivariate normal with density quadratic term Ξ 1 (a-2 px...
Suppose that X and Y are bivariate normal with density quadratic term Ξ 1 (a-2 px yty xor f(x,y) = This means that X and Y are correlated standard normal random variables since We will show that X and the new random variable Z defined as Since Z is obtained as a linear combination of normal random variables, it is also a. What is the mean of Z, call it E[Z]? b. What is the variance-covariance matrix of the random...
3. Suppose X,,X2,, is a random sample from a standard normal distribution and let Z be...
3. Suppose X,,X2,, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X,X, X, UX , and V X- 9 9 Let X (x - x)2 i-1 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 32 VU Determine the distribution of the variable (c) (d) (e) Determine the distribution of the variable y (where Y is...
Question 5 - Even More Fun With Bivariate Normal Distributions Let X and Y be independent normall...
Question 5 - Even More Fun With Bivariate Normal Distributions Let X and Y be independent normally distributed with mean x = 2 and μΥ--3 and standard deviations ơX-3 and ơY-5, respectively. Determine the following: (a) P(3X 6Y>15), (b) P(3X6Y<30) (c) Cov(X, Y) d) Verify (a) and (b) using R code, where for each case you generate a million X's and a million Y's and simulate the linear combination 3X 6Y. (e) Assume now that the random variables come from...
Suppose X1,X2, .. ,X, is a random sample from a standard normal distribution and let Z...
Suppose X1,X2, .. ,X, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X1, X2, .., X,. 9 9 9 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 3Z NU Determine the distribution of the variable W- (c) (d) Determine the distribution of the variable R -4y (where Y is the variable from (C)
4. Let X and Y be independent standard normal random variables. The pair (X,Y) can be...
4. Let X and Y be independent standard normal random variables. The pair (X,Y) can be described in polar coordinates in terms of random variables R 2 0 and 0 e [0,27], so that X = R cos θ, Y = R sin θ. (a) (10 points) Show that θ is uniformly distributed in [0,2 and that R and 0 are independent. (b) (IO points) Show that R2 has an exponential distribution with parameter 1/2. , that R has the...
X and Ý are both standard normal distributions, and their covariance is ε, which only takes two possible values, -1 and 1. Is the pair (X, Y) bivariate normal? X and Ý are both standard normal d...
X and Ý are both standard normal distributions, and their covariance is ε, which only takes two possible values, -1 and 1. Is the pair (X, Y) bivariate normal?
X and Ý are both standard normal distributions, and their covariance is ε, which only takes two possible values, -1 and 1. Is the pair (X, Y) bivariate normal?
9. Let X and Y be two random variables. Suppose that σ = 4, and σ...
9. Let X and Y be two random variables. Suppose that σ = 4, and σ -9. If we know that the two random variables Z-2X?Y and W = X + Y are independent, find Cov(X, Y) and ρ(X,Y). 10. Let X and Y be bivariate normal random variables with parameters μェー0, σ, 1,Hy- 1, ơv = 2, and ρ = _ .5. Find P(X + 2Y < 3) . Find Cov(X-Y, X + 2Y) 11. Let X and Y...
xercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < 1 . Check that if X Z and Y-: ρΖ+ VI-P" W then the pair (X, Y) has standard bivariate normal distribution with parameter p. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.
Exercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < l. Check that if X-Z and Y-p2+ VI-p-W then the pair (X, Y) has standard bivariate normal distribution with parameter ρ. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.
Problem 1. Let X be a normal random variable with mean 0 and variance 1 and let Y be uniform(0.1) with X and Y being independent. Let U-X + Y and V = X-Y. For this problem recall the density for a normal random variable is 2πσ2 (a) Find the joint distribution of U and V (b) Find the marginal distributions of U and V (c) Find Cov(U, V).
Suppose that X and Y are bivariate normal with density quadratic term Ξ 1 (a-2 px yty xor f(x,y) = This means that X and Y are correlated standard normal random variables since We will show that X and the new random variable Z defined as Since Z is obtained as a linear combination of normal random variables, it is also a. What is the mean of Z, call it E[Z]? b. What is the variance-covariance matrix of the random...
3. Suppose X,,X2,, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X,X, X, UX , and V X- 9 9 Let X (x - x)2 i-1 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 32 VU Determine the distribution of the variable (c) (d) (e) Determine the distribution of the variable y (where Y is...
Question 5 - Even More Fun With Bivariate Normal Distributions Let X and Y be independent normally distributed with mean x = 2 and μΥ--3 and standard deviations ơX-3 and ơY-5, respectively. Determine the following: (a) P(3X 6Y>15), (b) P(3X6Y<30) (c) Cov(X, Y) d) Verify (a) and (b) using R code, where for each case you generate a million X's and a million Y's and simulate the linear combination 3X 6Y. (e) Assume now that the random variables come from...
Suppose X1,X2, .. ,X, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X1, X2, .., X,. 9 9 9 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 3Z NU Determine the distribution of the variable W- (c) (d) Determine the distribution of the variable R -4y (where Y is the variable from (C)
4. Let X and Y be independent standard normal random variables. The pair (X,Y) can be described in polar coordinates in terms of random variables R 2 0 and 0 e [0,27], so that X = R cos θ, Y = R sin θ. (a) (10 points) Show that θ is uniformly distributed in [0,2 and that R and 0 are independent. (b) (IO points) Show that R2 has an exponential distribution with parameter 1/2. , that R has the...
X and Ý are both standard normal distributions, and their covariance is ε, which only takes two possible values, -1 and 1. Is the pair (X, Y) bivariate normal?
X and Ý are both standard normal distributions, and their covariance is ε, which only takes two possible values, -1 and 1. Is the pair (X, Y) bivariate normal?
9. Let X and Y be two random variables. Suppose that σ = 4, and σ -9. If we know that the two random variables Z-2X?Y and W = X + Y are independent, find Cov(X, Y) and ρ(X,Y). 10. Let X and Y be bivariate normal random variables with parameters μェー0, σ, 1,Hy- 1, ơv = 2, and ρ = _ .5. Find P(X + 2Y < 3) . Find Cov(X-Y, X + 2Y) 11. Let X and Y...
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