Question

1. (20 points) Write TRUE if the statement is true and write FALSE if the statement is false. (a) The quotient of any two chi-square distributed random variables divided by their respective degrees of freedom is F distributed. (b) The expected value of a sum of a finite number of random variables is the sum of the expected values of each random variable in the sum. (c) If X is uncorrelated with the random variables Y and Z, then Y and Z are uncorrelated. (d) The marginal distributions are uniquely determined from the joint distribution but knowledge of the marginal distributions is not sufficient to determine the joint distribution. (e) Let X = (X1, X2, ..., Xn)' be a random vector with joint cumulative distribution function fX(·). Then ? (Xi - ?i / ?i)^2 has a chi-square distribution with n degrees of freedom where ?i and ?i^2 are the mean and variance, respectively, of each of the random variable in X, i = 1, 2, ..., n. (f) Let X1, X2, ..., Xn be a random sample from a density f(·), which has mean ? and finite variance ?^2, and let X? = 1/n ? Xi. Then E[X?] = ? and Var(X?) = ?^2/n. (g) For any random variables X and Y, E[XY] = E[X] E[Y]. (h) Zero covariance or correlation is a sufficient condition for two random variables to be independent. (i) The random variables X1, X2, ..., Xk are independent if and only if m_{X1, X2, ..., Xk}(t1, t2, ..., tk) = ? m_{Xi}(ti). (j) Let X1 and X2 be two random variables, then ? a1, a2 ? ?, Var(a1X1 - a2X2) = a1^2 Var(X1) + a2^2 Var(X2) - 2a1a2 Cov(X1, X2).

          1. (20 points) Write TRUE if the statement is true and write FALSE if the statement is false.
(a) The quotient of any two chi-square distributed random variables divided by their respective degrees of freedom is F distributed.
(b) The expected value of a sum of a finite number of random variables is the sum of the expected values of each random variable in the sum.
(c) If X is uncorrelated with the random variables Y and Z, then Y and Z are uncorrelated.
(d) The marginal distributions are uniquely determined from the joint distribution but knowledge of the marginal distributions is not sufficient to determine the joint distribution.
(e) Let X = (X1, X2, ..., Xn)' be a random vector with joint cumulative distribution function fX(·). Then ? (Xi - ?i / ?i)^2 has a chi-square distribution with n degrees of freedom where ?i and ?i^2 are the mean and variance, respectively, of each of the random variable in X, i = 1, 2, ..., n.
(f) Let X1, X2, ..., Xn be a random sample from a density f(·), which has mean ? and finite variance ?^2, and let X? = 1/n ? Xi. Then E[X?] = ? and Var(X?) = ?^2/n.
(g) For any random variables X and Y, E[XY] = E[X] E[Y].
(h) Zero covariance or correlation is a sufficient condition for two random variables to be independent.
(i) The random variables X1, X2, ..., Xk are independent if and only if m_{X1, X2, ..., Xk}(t1, t2, ..., tk) = ? m_{Xi}(ti).
(j) Let X1 and X2 be two random variables, then ? a1, a2 ? ?, Var(a1X1 - a2X2) = a1^2 Var(X1) + a2^2 Var(X2) - 2a1a2 Cov(X1, X2).

1. (20 points) Write TRUE if the statement is true and write FALSE if the statement is false.
(a) The quotient of any two chi-square distributed random variables divided by their respective degrees of freedom is F distributed.
(b) The expected value of a sum of a finite number of random variables is the sum of the expected values of each random variable in the sum.
(c) If X is uncorrelated with the random variables Y and Z, then Y and Z are uncorrelated.
(d) The marginal distributions are uniquely determined from the joint distribution but knowledge of the marginal distributions is not sufficient to determine the joint distribution.
(e) Let X = (X1, X2, ..., Xn)' be a random vector with joint cumulative distribution function fX(·). Then ? (Xi - ?i / ?i)^2 has a chi-square distribution with n degrees of freedom where ?i and ?i^2 are the mean and variance, respectively, of each of the random variable in X, i = 1, 2, ..., n.
(f) Let X1, X2, ..., Xn be a random sample from a density f(·), which has mean ? and finite variance ?^2, and let X? = 1/n ? Xi. Then E[X?] = ? and Var(X?) = ?^2/n.
(g) For any random variables X and Y, E[XY] = E[X] E[Y].
(h) Zero covariance or correlation is a sufficient condition for two random variables to be independent.
(i) The random variables X1, X2, ..., Xk are independent if and only if mX1, X2, ..., Xk(t1, t2, ..., tk) = ? mXi(ti).
(j) Let X1 and X2 be two random variables, then ? a1, a2 ? ?, Var(a1X1 - a2X2) = a1^2 Var(X1) + a2^2 Var(X2) - 2a1a2 Cov(X1, X2).

Added by Whitney N.

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