# Probability & Statistics

• November 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.

• November 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 ${\color{red} \frac{2 (\theta - x}{\theta^2}}$.
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 $\theta$ using the given pdf.

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

Now show that $E(\hat{\theta}) = \theta$.
• November 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
• November 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
• November 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

$E(X) = \int_0^{\theta} x \, \frac{2 (\theta - x)}{\theta^2} \, dx = \frac{2}{\theta^2} \int_0^{\theta} x \, (\theta - x) \, dx$.
• November 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)?
• November 30th 2008, 01:55 AM
James1220
Ooo never mind! I've found it :) thanks!
• December 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!
• December 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!

Now try the question again.

Post if you're still stuck (please be sure to say where you're stuck).