# estimation of parameter

• Nov 12th 2010, 01:35 PM
wik_chick88
estimation of parameter
consider the probability mass function
$p_{X} (x; \theta) = \theta ^{x} (1 - \theta )^{1-x}, x = 0,1$
and suppose we observe a sample of 5 outcomes: 0, 0, 1, 0, 1.
(a) show how maximum likelihood estimation can be used to estimate the parameter $\theta$
(b) show how the method of moments can be used to estimate the parameter $\theta$
• Nov 12th 2010, 09:53 PM
CaptainBlack
Quote:

Originally Posted by wik_chick88
consider the probability mass function
$p_{X} (x; \theta) = \theta ^{x} (1 - \theta )^{1-x}, x = 0,1$
and suppose we observe a sample of 5 outcomes: 0, 0, 1, 0, 1.
(a) show how maximum likelihood estimation can be used to estimate the parameter $\theta$
(b) show how the method of moments can be used to estimate the parameter $\theta$

(a) Well what is the likelihood of exactly 2 1'1 in 5 trials?

(b) Set the expected number of 1's in 5 trials to the observed result.

CB
• Nov 27th 2010, 11:23 PM
wik_chick88
im still confused on how to start this question?
• Nov 28th 2010, 04:19 AM
CaptainBlack
Quote:

Originally Posted by CaptainBlack
(a) Well what is the likelihood of exactly 2 1'1 in 5 trials?

Consider the 1's as successes then and $N$ the number of successes in 5 trials:

$L(\theta|N=2)=k \theta^N (1-\theta)^{5-N}=k \theta^2 (1-\theta)^{3}$

Then the maximum liklyhood estimator is the largest $\Theta$ which solves:

$\dfrac{d}{d\theta}L(\theta|N=2)=0$

CB
• Dec 15th 2010, 04:54 AM
wik_chick88
i got $\theta = \frac{2}{5}$ is this correct?
also, im still confused as what to do for (b), how do i calculate the expected number of 1's in 6 trials?
thanks so much!
• Dec 15th 2010, 05:21 AM
CaptainBlack
Quote:

Originally Posted by wik_chick88
i got $\theta = \frac{2}{5}$ is this correct?

Yes
• Dec 15th 2010, 05:29 AM
CaptainBlack
Quote:

Originally Posted by wik_chick88
iim still confused as what to do for (b), how do i calculate the expected number of 1's in 6 trials?
thanks so much!

We are setting $p_X(1;\theta)=\theta=2/5$

(we are setting the mean of X to the observed proportion in our sample, which is matching first monents of the distribution of number of 1's in a sample with the observed)

CB
• Dec 15th 2010, 06:04 AM
wik_chick88
im confused! how do i know (or guess) whether the 6th outcome will be a 0 or a 1?
• Dec 15th 2010, 11:17 AM
CaptainBlack
Quote:

Originally Posted by wik_chick88
im confused! how do i know (or guess) whether the 6th outcome will be a 0 or a 1?

The 6 is a typo for 5

CB