Answer: Suppose that X1,...,Xn form a random sample from a

Let x1,...,xn be distinct numbers. Let Y be a discrete random variable with the following p.f.: f (y) =  1 n if y {x1,...,xn}, 0 otherwise. Prove that Var(Y ) is given by Eq. (7.5.5).

It is not known what proportion p of the purchases of a certain brand of breakfast cereal are made by women and what proportion are made by men. In a random sample of 70 purchases of this cereal, it was found that 58 were made by women and 12 were made by men. Find theM.L.E. of p

. Consider again the conditions in Exercise 2, but suppose also that it is known that 1 2 p 2 3 . If the observations in the random sample of 70 purchases are as given in Exercise 2, what is the M.L.E. of p?

Suppose that X1,...,Xn form a random sample from the Bernoulli distribution with parameter , which is unknown, but it is known that lies in the open interval 0 << 1. Show that the M.L.E. of does not exist if every observed value is 0 or if every observed value is 1

Suppose that X1,...,Xn form a random sample from a Poisson distribution for which the mean is unknown, ( > 0). a. Determine the M.L.E. of , assuming that at least one of the observed values is different from 0. b. Show that the M.L.E. of does not exist if every observed value is 0.

Suppose that X1,...,Xn form a random sample from a normal distribution for which the mean is known, but the variance 2 is unknown. Find the M.L.E. of 2.

Suppose that X1,...,Xn form a random sample from an exponential distribution for which the value of the parameter is unknown ( > 0). Find the M.L.E. of

Suppose that X1,...,Xn form a random sample from a distribution for which the p.d.f. f (x| ) is as follows: f (x| ) =  ex for x>, 0 for x . Also, suppose that the value of is unknown ( << ). a. Show that the M.L.E. of does not exist. b. Determine another version of the p.d.f. of this same distribution for which the M.L.E. of will exist, and find this estimator.

uppose that X1,...,Xn form a random sample from a distribution for which the p.d.f. f (x| ) is as follows: f (x| ) =  x1 for 0 0). Find the M.L.E. of .

Suppose that X1,...,Xn form a random sample from a distribution for which the p.d.f. f (x| ) is as follows: f (x| ) = 1 2 e|x| for

Suppose that X1,...,Xn form a random sample from the uniform distribution on the interval [1, 2], where both 1 and 2 are unknown ( < 1 < 2 < ). Find the M.L.E.s of 1 and 2

Suppose that a certain large population contains k different types of individuals (k 2), and let i denote the proportion of individuals of type i, for i = 1,...,k. Here, 0 i 1 and 1 + ... + k = 1. Suppose also that in a random sample of n individuals from this population, exactly ni individuals are of type i, where n1 + ... + nk = n. Find the M.L.E.s of 1,...,k.

Suppose that the two-dimensional vectors (X1, Y1), (X2, Y2), . . . , (Xn, Yn) form a random sample from a bivariate normal distribution for which the means of X and Y are unknown but the variances of X and Y and the correlation between X and Y are known. Find the M.L.E.s of the mean