1.)  Maximum likelihood estimation.  In this problem, Y=B+1 with probability v, and Y=B with probability 1-v, where v is known. You must estimate B given Y.

a.   Show that the likelihood function is given by fY | B (y|B)=1-v, for B=y, and fY | B (y|B)=v for B=y-1.

b.   For v=1/4, find the maximum likelihood estimator for B using Y=y.

c.   Suppose that {Nj } j=1,..D is an independent set of identically distributed Bernoulli trials, with P[Nj=1]=v, and P[Nj=0]=1-v.  Both D and v are known, and suppose       that Yj=B+Nj, j=1,…D.  Note that the same realization of B is used for all the Yj,       and that each Yj equals B or B+1 independently for each index j.  Show that Ymin      =min{Y1, Y2, Y3, …YD } is sufficient to estimate B given the observation Y1, Y2, Y3,      …YD .  (A statistic t(Y1, Y2, Y3, …YD ) is sufficient for B if the likelihood function of B

for the measurements Y1, Y2, Y3, …YD depends on the measurements only through t.)

d.   The likelihood function for B given Ymin  is f(ymin |B)=1-vD, for ymin=B, and                  f(ymin |B)=vD, for ymin=B+1.  Find the maximum likelihood estimator for B given Y1,

Y2, …YD .

2. Bayesian estimation.  Suppose {Yj } are defined as in problem 1.  Additionally, it is known that parameter B has a probability density function fB (b)=e-b, b>=0.

a.   Using the prior density of B only, what is an estimator of B?  Provide an explanation of your choice of estimator.

b.   Using the given information in 1.d, as well as the prior distribution, what is the                maximum a posteriori estimator of B given {Y1, Y2,… YD }?  Explain your answer carefully, providing all calculations.