# Thread: Continuous random variables

1. ## Continuous random variables

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.

2. Originally Posted by bill2010
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.
I think we need more information to get any useful answer. Are these sequence of random variables i.i.d?

3. 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.

4. Originally Posted by bill2010
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.
Originally Posted by bill2010
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:

$\displaystyle E\left( \frac{X_1 + X_2 + \, .... \, + X_n}{n - 1} \right) = \frac{1}{n-1} E(X_1) + \frac{1}{n-1} E(X_2) + \, .... \, + \frac{1}{n-1} E(X_n)$

$\displaystyle = \frac{1}{n-1} \left (E(X_1) + E(X_2) + \, .... + E(X_n) \right) = \frac{1}{n-1} (\mu + \mu + \, .... + \mu)$

$\displaystyle = \frac{n \mu}{n-1}$.

$\displaystyle Var\left( \frac{X_1 + X_2 + \, .... \, + X_n}{n - 1} \right)$

$\displaystyle = \frac{1}{(n-1)^2} Var(X_1) + \frac{1}{(n-1)^2}Var(X_2) + \, .... \, + \frac{1}{(n-1)^2} Var(X_n)$

$\displaystyle = \frac{1}{(n-1)^2} \left (Var(X_1) + Var(X_2) + \, .... + Var(X_n) \right) = \frac{1}{(n-1)^2} (\sigma^2 + \sigma^2 + \, .... + \sigma^2)$

$\displaystyle = \frac{n \sigma^2 }{(n-1)^2}$.

Originally Posted by bill2010
[snip]i have to state which is the best and worst estimator out of the 3 given[snip]
You don't say "best and worst" in which sense ....