Im stuck with the following -
(Xn-1 + Xn)/2;
(1/4)X1 + (3/4)X2;
(X1 + X2 + ……… + Xn)/(n – 1)
I need to derive expressions for the expected values and variances of the (three) above estimators of mx.
Any help greatly appreciated.
Im stuck with the following -
(Xn-1 + Xn)/2;
(1/4)X1 + (3/4)X2;
(X1 + X2 + ……… + Xn)/(n – 1)
I need to derive expressions for the expected values and variances of the (three) above estimators of mx.
Any help greatly appreciated.
X is a continuous random variable, and n is the sample size taken from the population.
It is trying to derive the mean and variance, i have to state which is the best and worst estimator out of the 3 given.
That is all the information in the question im afriad.
You need to apply the following formulae:
1. $\displaystyle E(aX_1 + bX_2 + \, .... ) = a E(X_1) + b E(X_2) + \, ....$
2. $\displaystyle Var(aX_1 + bX_2 + \, .... ) = a^2 Var(X_1) + b^2 Var(X_2) + \, ....$
when $\displaystyle X_1, \, X_2, \, ....$ are independent (which they are here).
For example:
$\displaystyle E\left( \frac{X_1 + X_2 + \, .... \, + X_n}{n - 1} \right) = \frac{1}{n-1} E(X_1) + \frac{1}{n-1} E(X_2) + \, .... \, + \frac{1}{n-1} E(X_n)$
$\displaystyle = \frac{1}{n-1} \left (E(X_1) + E(X_2) + \, .... + E(X_n) \right) = \frac{1}{n-1} (\mu + \mu + \, .... + \mu)$
$\displaystyle = \frac{n \mu}{n-1}$.
$\displaystyle Var\left( \frac{X_1 + X_2 + \, .... \, + X_n}{n - 1} \right)$
$\displaystyle = \frac{1}{(n-1)^2} Var(X_1) + \frac{1}{(n-1)^2}Var(X_2) + \, .... \, + \frac{1}{(n-1)^2} Var(X_n)$
$\displaystyle = \frac{1}{(n-1)^2} \left (Var(X_1) + Var(X_2) + \, .... + Var(X_n) \right) = \frac{1}{(n-1)^2} (\sigma^2 + \sigma^2 + \, .... + \sigma^2)$
$\displaystyle = \frac{n \sigma^2 }{(n-1)^2}$.
You don't say "best and worst" in which sense ....