Assume that the random variables X1 and X2 are bivariate normally distributed with mean µ and covariance matrix Σ. Show that if cov(X1, X2) = 0 then X1 and X2 are independent.
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Assume that the random variables X1 and X2 are bivariate
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- Let X1, X2, X3 and X4 be exponential(1) random variables. Find the joint distribution of X1/(X1+X2+X3+X4) and (X1+X2)/(X1+X2+X3+X4) and (X1+X2+X3)/(X1+X2+X3+X4) using Jacobian method?Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If B is the linear minimum variance estimate of B based on y, show that the covariance of the error B - B has rank n - m.If X is a random variable having the standard normaldistribution and Y = X2, show that cov(X, Y) = 0 eventhough X and Y are evidently not independent.
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