Consider three random variables X1, X2 and X3. For all three variables the expectations are equal...

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Consider three random variables X1, X2 and X3. For all three variables the expectations are equal...

Consider three random variables X1, X2 and X3. For all three variables the expectations are equal to µ and the variances are equal to ν, also, the correlation between any two of them is equal to r.

(a) Find the correlation between U = X1 − X2 and V = X2 − X3.

(b) Find E[U(U − 2V )] (hint: use alternative definition of covariance).

(c) Use Jensen’s inequality with the transformation g(t) = (e^t)^2 to find a lower bound for E[(e^U)^2].

(d) Use Jensen’s inequality with the transformation h(t) = e^t to find a lower bound for E[(e^U)^2].

(e) Comment on the efficacy of the two bounds.

math Statistics-And-Probability

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