# Probability & Statistics

• Nov 27th 2008, 07:45 AM
James1220
Probability & Statistics
A random sample of two functions x1, x2, is drawn from the distribution with pdf:

2/θ^2 (θ - x) 0<x<θ
f(x;θ) = {
0 otherwise

find the method of moments estimator of θ and show that it is not bias.

• Nov 27th 2008, 10:28 PM
mr fantastic
Quote:

Originally Posted by James1220
A random sample of two functions x1, x2, is drawn from the distribution with pdf:

2/θ^2 (θ - x) 0<x<θ Mr F says: I assume you mean $\displaystyle {\color{red} \frac{2 (\theta - x}{\theta^2}}$.
f(x;θ) = {
0 otherwise

find the method of moments estimator of θ and show that it is not bias.

Calculate E(X) as a function of $\displaystyle \theta$ using the given pdf.

Note that the sample mean is $\displaystyle \bar{X} = \frac{x_1 + x_2}{2}$. Estimate $\displaystyle \theta$ by equating E(X) to the sample mean. I get $\displaystyle \hat{\theta} = \frac{3(x_1 + x_2)}{2} = 3 \bar{X}$.

Now show that $\displaystyle E(\hat{\theta}) = \theta$.
• Nov 29th 2008, 04:47 AM
James1220
Do i get the mean E(X) by integrating? Im not managing to get it so far. I got 1 - (2x)/theta
• Nov 29th 2008, 06:28 AM
selinunan
You should take this integral:

E(x)=∫xf(x)dx=∫x[(2/(Q²)(Q-x)]dx

The result is E(x)=Q/3
• Nov 29th 2008, 12:25 PM
mr fantastic
Quote:

Originally Posted by James1220
Do i get the mean E(X) by integrating? Im not managing to get it so far. I got 1 - (2x)/theta

$\displaystyle E(X) = \int_0^{\theta} x \, \frac{2 (\theta - x)}{\theta^2} \, dx = \frac{2}{\theta^2} \int_0^{\theta} x \, (\theta - x) \, dx$.
• Nov 30th 2008, 01:49 AM
James1220
Thank you Mr F and selinunan. I have finally integrated and found E(X)! And yes it is θ/3. When showing that it is unbiased, theta hat = 3 xbar. How do i go about finding E(theta hat)?
• Nov 30th 2008, 01:55 AM
James1220
Ooo never mind! I've found it :) thanks!
• Dec 2nd 2008, 04:57 AM
James1220
Quote:

Originally Posted by James1220
Ooo never mind! I've found it :) thanks!

A random sample of two functions x1, x2, is drawn from the distribution with pdf:

2/θ^2 (θ - x) 0<x<θ
f(x;θ) = {
0 otherwise

find the method of moments estimator of θ and show that it is not bias.

With help of fellow mathematicians, i found the solution to the above problem. the following question relates to this and is the second part of the question :

Show that the maximum likelihood estimator of θ is given by:

\hat{\theta} = 1/4 [3(x_1 + x_2) + Square root of (9x_1^2 - 14x_1x_2 + 9x_2^2)]

Im troubled because of the fact that there are 2 variables x_1 and x_2, so i'm having trouble finding how to fond the mle. Help if you can Pleaseee!
• Dec 2nd 2008, 02:12 PM
mr fantastic
Quote:

Originally Posted by James1220
A random sample of two functions x1, x2, is drawn from the distribution with pdf:

2/θ^2 (θ - x) 0<x<θ
f(x;θ) = {
0 otherwise

find the method of moments estimator of θ and show that it is not bias.

With help of fellow mathematicians, i found the solution to the above problem. the following question relates to this and is the second part of the question :

Show that the maximum likelihood estimator of θ is given by:

\hat{\theta} = 1/4 [3(x_1 + x_2) + Square root of (9x_1^2 - 14x_1x_2 + 9x_2^2)]

Im troubled because of the fact that there are 2 variables x_1 and x_2, so i'm having trouble finding how to fond the mle. Help if you can Pleaseee!