Suppose Z1 and Z2 are two standard norm random variables. In addition suppose cov(Z1,Z2)=p. Show (Z1-pZ2)/sqrt(1-p^2)and...

Suppose Z1 and Z2 are two standard norm random variables. In addition suppose cov(Z1,Z2)=p.

Show (Z1-pZ2)/sqrt(1-p^2)and Z2 are standard normally distributed
Show (Z1-pZ2) )/sqrt(1-p^2)and Z2 are independent. Hint : Two random normal variables are independent as long as they are uncorrelated
Show (Z1^2+Z2^2-2pZ1Z2)/(1-p^2) is Chai square distribution

Expert Solution

orchestra answered 1 year ago

1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2,...

1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2, Y2 = Z1 − Z2. (a) Find the joint pdf fY1,Y2(y1,y2). Don’t use the change of variables theorem – all of that work has already been done for you. Instead, evaluate the matrices Σ and Σ−1, then multiply the necessary matrices and vectors to obtain a formula for fY1,Y2(y1,y2) containing no matrices and no vectors. (b) Find the marginal pdf fY2 (y2). Don’t use...

Provide an example that if the cov(X,Y) = 0, the two random variables, X and Y,...

Provide an example that if the cov(X,Y) = 0, the two random variables, X and Y, are not necessarily independent. Would you please give the example specifically and why?

Provide an example that if the cov(X,Y) = 0, the two random variables, X and Y,...

Provide an example that if the cov(X,Y) = 0, the two random variables, X and Y, are not necessarily independent. Would you please give the example specifically and why?

A: Suppose two random variables X and Y are independent and identically distributed as standard normal....

A: Suppose two random variables X and Y are independent and identically distributed as standard normal. Specify the joint probability density function f(x, y) of X and Y. Next, suppose two random variables X and Y are independent and identically distributed as Bernoulli with parameter 1 2 . Specify the joint probability mass function f(x, y) of X and Y. B: Consider a time series realization X = [10, 15, 23, 20, 19] with a length of five-periods. Compute the...

Suppose that X1 and X2 are two random variables. Suppose that X1 has mean 1 and...

Suppose that X1 and X2 are two random variables. Suppose that X1 has mean 1 and variance 4 while X2 has mean 3 and variance 9. Finally, suppose that the correlation between X1 and X2 is 3/8. Denote Y = 2X1 − X2. (67) The mean of Y is (a) 1 (b) 4 (c) -2 (d) -1 (68) The variance of Y is (a) 25 (b) 4 (c) 9 (d) 16 (69) The standard deviation of Y is (a) 5...

Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,...

Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1, P(X = 1, Y = 0) = .3, P(X = 2, Y = 0) = .2 P(X = 0, Y = 1) = .2, P(X = 1, Y = 1) = .2, P(X = 2, Y = 1) = 0. a. Determine E(X) and E(Y ). b. Find Cov(X, Y ) c. Find Cov(2X + 3Y, Y ).

Suppose Z denotes the standard normal random variable. Compute the following a) P (Z > 2)...

Suppose Z denotes the standard normal random variable. Compute the following a) P (Z > 2) b) P(-1 < Z < 2.31) c) P (1.21 < Z < 2.42) d) P (Z < 1.37) e) P (Z > -1) Show working

Suppose that X and Y are two normally distributed random variables. X has mean 2 and...

Suppose that X and Y are two normally distributed random variables. X has mean 2 and standard deviation 3.Y has mean 3 and standard deviation 2. Their correlation is 0.6 . What is the mean and standard deviation of X + Y ?What is the distribution of X + Y ? What if X and Y are jointly normally distributed? What if they are not jointly normally distributed? Explain your answer.

Suppose that X and Y are two normally distributed random variables. X has mean 2 and...

Suppose that X and Y are two normally distributed random variables. X has mean 2 and standard deviation 3.Y has mean 3 and standard deviation 2. Their correlation is 0.6 . What is the mean and standard deviation of X + Y? What is the distribution of X + Y? What if X and Y are jointly normally distributed? What if they are not jointly normally distributed? Explain your answer. (Why we cannot conclude if they are not jointly normally...

Suppose that a firm has the p production function f(x1; x2) = sqrt(x1) + x2^2. (a)...

Suppose that a firm has the p production function f(x1; x2) = sqrt(x1) + x2^2. (a) The marginal product of factor 1 (increases, decreases, stays constant) ------------ as the amount of factor 1 increases. The marginal product of factor 2 (increases, decreases, stays constant) ----------- as the amount of factor 2 increases. (b) This production function does not satisfy the definition of increasing returns to scale, constant returns to scale, or decreasing returns to scale. How can this be? (c)Find...

Question

Suppose Z1 and Z2 are two standard norm random variables. In addition suppose cov(Z1,Z2)=p. Show (Z1-pZ2)/sqrt(1-p^2)and...

Solutions

Expert Solution

Related Solutions

1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2,...

Provide an example that if the cov(X,Y) = 0, the two random variables, X and Y,...

Provide an example that if the cov(X,Y) = 0, the two random variables, X and Y,...

A: Suppose two random variables X and Y are independent and identically distributed as standard normal....

Suppose that X1 and X2 are two random variables. Suppose that X1 has mean 1 and...

Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,...

Suppose Z denotes the standard normal random variable. Compute the following a) P (Z > 2)...

Suppose that X and Y are two normally distributed random variables. X has mean 2 and...

Suppose that X and Y are two normally distributed random variables. X has mean 2 and...

Suppose that a firm has the p production function f(x1; x2) = sqrt(x1) + x2^2. (a)...