Finding mean and variance from the mean?

wolverine21 · #1 $Old$ November 18th, 2008, 01:19 AM

Find the mean and variance of the mean of a random sample of 9 from a distribution having pdf 6x(1-x) 0 < x < 1, zero elsewhere.

I don't even know how to start this one.

mr fantastic · #2 $Old$ November 18th, 2008, 04:56 AM

Quote:

Originally Posted by wolverine21 $View Post$

Find the mean and variance of the mean of a random sample of 9 from a distribution having pdf 6x(1-x) 0 < x < 1, zero elsewhere.

I don't even know how to start this one.

Start with the basic definitions and formulae:

$\bar{X} = \frac{X_1 + X_2 + \, .... \, + X_9}{9}$ .

$E(\bar{X}) = \frac{1}{9} \left[ E(X_1) + E(X_2) + \, .... \, + E(X_9) \right] = E(X_i)$

where $E(X_i) = \int_0^1 6x^2 (1 - x) \, dx$ .

$Var(\bar{X}) = \frac{1}{81} \left[ Var(X_1) + Var(X_2) + \, .... \, + Var(X_9) \right] = \frac{1}{9} Var(X_i)$

where $Var(X_i) = E(X_i^2) - [E(X_i)]^2$ and $E(X_i^2) = \int_0^1 6x^3 (1 - x) \, dx$ .