Homework 8. More on maximal likelihood. Bayesian
Due Thursday, 11/03, by 11 am
(1) Let X1 , X2 , . . . , Xn be i.i.d. random variables, Xi ∼ Geometric(p).
(a) Find the MLE pb of p.
(b) Is the bias E[b
p] − p positive, negative, or zero for n = 1 and 0 < p < 1? (2) Twin pairs are either homozygotic (with probability α) or dizygotic (probability 1 − α). In the first case, their common sex is genetically determined only once, for both twins together, so that they must necessarily be of the same sex. In the second case, the sexes are determined independently and therefore could potentially be different. Suppose that generally in any genetic sex determination in a given population the outcome will be “male” (m) with probability β and “female” (f) with probability 1 − β. Obviously, the sex-related classification of any given twin pair (i.e., mm vs. ff vs. mf) is readily apparent, whereas the determination of their homo- vs. dizygotic status requires elaborate genetic diagnostics. Specifically, if the two twins are of opposite sexes (i.e., the pair is mf) then they must necessarily be dizygotic, but if they are not (i.e., an mm or ff pair) then they may be either homo- or dizygotic. L. v. Bortkiewicz (1920) has collected the sexrelated classification for a total of n = 17798 twin pairs born in Berlin from 1879 to 1911. The frequencies were n(mm) = 5844, n(f f ) = 5612, and n(mf ) = 6342. (a) Use maximal likelihood method to find estimates of pmm and pf f . Write down the corresponding equations. You can use any software to solve them numerically. (b) Find maximal likelihood estimates of α and β. (3) Consider two boxes, one containing 2 black and 2 white marbles, the other 3 black and 5 white marbles. A box is selected at random, and a marble is drawn from it at random. What is the probability that the marble is black? What is the probability that the first box was the one selected given that the marble is white? (4) Suppose that a statistician believes that probability p to be cured by a new drug is beta with mean 0.8 and variance 0.01. In a trial with 10 patients 7 people were cured. Find mean and variance of the posterior distribution of p. (5) X1 , X2 , . . . , Xn are i.i.d. random variables, Xi has probability density function f (x, a) = ax−a−1 for 1 ≤ x < ∞. Prior distribution of a is exponential with parameter 1. Find mean of the posterior distribution of a. 1
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