Thread: 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.


Please help!

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 .
f(x;θ) = {
0 otherwise

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


Please help!

Calculate E(X) as a function of using the given pdf.

Note that the sample mean is . Estimate by equating E(X) to the sample mean. I get .

Now show that .

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

You should take this integral:

E(x)=∫xf(x)dx=∫x[(2/(Q²)(Q-x)]dx

The result is E(x)=Q/3

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

.

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)?

Ooo never mind! I've found it thanks!

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!

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!

Read this thread carefully: http://www.mathhelpforum.com/math-he...estimator.html

Now try the question again.

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