1. ## estimation of parameter

consider the probability mass function
$\displaystyle p_{X} (x; \theta) = \theta ^{x} (1 - \theta )^{1-x}, x = 0,1$
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 $\displaystyle \theta$
(b) show how the method of moments can be used to estimate the parameter $\displaystyle \theta$

2. Originally Posted by wik_chick88
consider the probability mass function
$\displaystyle p_{X} (x; \theta) = \theta ^{x} (1 - \theta )^{1-x}, x = 0,1$
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 $\displaystyle \theta$
(b) show how the method of moments can be used to estimate the parameter $\displaystyle \theta$
(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

3. im still confused on how to start this question?

4. 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 $\displaystyle $$N the number of successes in 5 trials: \displaystyle L(\theta|N=2)=k \theta^N (1-\theta)^{5-N}=k \theta^2 (1-\theta)^{3} Then the maximum liklyhood estimator is the largest \displaystyle$$\Theta$ which solves:

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

CB

5. i got $\displaystyle \theta = \frac{2}{5}$ 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!

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

7. 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 $\displaystyle p_X(1;\theta)=\theta=2/5$

(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

8. im confused! how do i know (or guess) whether the 6th outcome will be a 0 or a 1?

9. 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