final_true_or_false.docx - (a If random variables X and Y...

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(a) If random variables X and Y are uncorrelated (i.e. Cor(X, Y ) = 0), then X and Y must be independent. If independent, p(x intersect y) = p(x)*p(y), cor = E(XY)-E(X)*E(Y) = integral xy * p(x, y) – integral x*p(x) * y*p(y) = 0 C.f. reading 3-17 (b) If P(A) < P(B) and P(C) > 0, then P(A|C) ≤ P(B|C). (c) The calculation of p-value does not depend on the alternative hypothesis once we know the null hypothesis. (d) The maximum likelihood estimator (MLE) is always unbiased. A homework problem with 2 data for each set has bias that does not converge to 0 as n increases. Generally estimators may be slightly biased. (e) For any hypothesis testing procedure, it is always possible to increase the power π = 1 β while keeping the type 1 error α the same. With likelihood ratio test being maximum (f) If X is a continuous random variable with cdf F(X), and Y is a random variable such that Y = F (X ), then Y is distributed uniformly on [0, 1]. Page 11 of Mar 2 lecture (g) Γ(1/2) = sqrt(pi) > Γ(3/2) = .5*sqrt(pi) < Γ(5/2) = .75*sqrt(pi) < Γ(7/2) = 1.875*sqrt(pi)
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