Statistics and Probability Answers

X1 , X2 and X3 is a random sample of size 3 from a population with mean mu and
variance sigma^2. T1, T2 and T3 are the estimators to estimate mu, and are given by
T1 = X1 + X2 - X3; T2 = 2X1 + 3X3 - 4X2 and T3 = 1/3(lambda X1 + X2 + X3).
(i) Are T 1 and T2 unbiased ? Give reason.
(ii) Find the value of A, such that T3 is
unbiased.
(iii) Which is the best estimator ? State
giving reasons.

X is a random variable taking values 0 and 1 with respective probabilities q and p. A sample X1, X2, ..., Xn of size n is taken from the distribution. If r = summation i = 1 to n(Xi) , show that (r + 1))/(n + 1) is a biased estimator of p. Also show that the bias of the estimator tends to zero
as n tends to infinity.

Let X be a random variable with pdf
f( x) = theta e ^(- theta x); theta > 0, x greater than equal to 0.
Find the moment generating function of X,
and hence find first three moments about
origin.

A random sample of size n is drawn from a uniform population over [ theta -1/ 2, theta + 1/2]
Obtain maximum likelihood estimator of theta. Does a unique estimator exist ? Give
reasons.

True or false.
X1, X2, ... Xn is a random sample from a uniform distribution with probability density function
f(x) = 1 / b-a, a < x < b , 0,otherwise
Then the maximum likelihood estimator of a is min (X1, X2, ... Xn).

Show that the maximum likelihood estimator of the parameter theta of a population having the density function (2/ theta^2) (theta - x), 0 < x < theta, for a sample of unit size is 2x, where x is the sample value. Also, show that the estimate obtained is biased.

True or false ?
1. If X1, X2, ..., XE is a random sample, then
the sample mean is an unbiased estimator of
the population mean.

2. A maximum likelihood estimator is always
unbiased.

T is an unbiased estimator for theta, show that
T^2 is a biased estimator for theta^2 .

Let X1, X2, ..., X, be a random sample from the following probability density function :
f(x, 0) = theta e^-theta x; x is greater than equal to 0, theta > 0
Estimate theta by the method of moments.