Let (X1, X2) be random variables. Determine whether (X1, X2) are independent in each of...
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Let (X1, X2) be random variables. Determine whether (X1, X2) are independent in each of the following examples:
f(x1, x2) = 12x1x2(1 x2) for x1 [0, 1], x2 [0, 1]; 0 elsewhere.
f(x1, x2) = e x1 e x2 for x1 > 0, x2 > 0; 0 elsewhere.
f(x1, x2) = 1/ for x12 + x22 1; 0 elsewhere.
MXY (t1, t2) = (1/(1t1)(1t2)) , t1 < 1, t2 < 1.
MXY (t1, t2) = exp(t1 + t2 + 0.5(t12 + t22 + 0.5t1t2)).
Hint: Think about factorization theorems. No need for complicated calculations!
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