# Thread: Consider a random sample of size n

1. ## Consider a random sample of size n

Question :

Consider a random sample of size $\displaystyle n$ from a population with the probability distribution $\displaystyle f(x,p)$ that depends on parameter $\displaystyle p$. Find the maximum likelihood estimator for $\displaystyle p$ when

$\displaystyle f(x,p) = p^x (1-p)^{1-x} \qquad for \ x = 0,1$

2. Originally Posted by zorro
Question :

Consider a random sample of size $\displaystyle n$ from a population with the probability distribution $\displaystyle f(x,p)$ that depends on parameter $\displaystyle p$. Find the maximum likelihood estimator for $\displaystyle p$ when

$\displaystyle f(x,p) = p^x (1-p)^{1-x} \qquad for \ x = 0,1$
Find the value of p that maximises L = f(0, p) f(1, p).

If you need more help, please show all of your working and state specifically where you're still stuck.

3. This is getting tiresome.
You posted 20 homework problems,
you don't make any effort
and on top of that you never thank anyone.
I'd ban this behaviour way before I'd ban someone from asking a question in the chatroom.

4. ## Is this correct?

Originally Posted by mr fantastic
Find the value of p that maximises L = f(0, p) f(1, p).

If you need more help, please show all of your working and state specifically where you're still stuck.

here is what i have done

$\displaystyle L(p)$ = $\displaystyle p^x (1 - p)^{1 - x}$

Taking log on both sides

$\displaystyle ln[L(p)]$ = $\displaystyle x \ ln(p) + (1 - x) \ ln(1 - p)$

Differentiating on both sides wrt p weget

$\displaystyle \frac{d}{dp} \ ln[L(p)]$ = $\displaystyle \frac{x}{p} + \frac{1 - x}{1 - p}$

Is it correct or have i done some thing wrong now what should i do further...

5. Originally Posted by zorro
here is what i have done

$\displaystyle L(p)$ = $\displaystyle p^x (1 - p)^{1 - x}$

Taking log on both sides

$\displaystyle ln[L(p)]$ = $\displaystyle x \ ln(p) + (1 - x) \ ln(1 - p)$

Differentiating on both sides wrt p weget

$\displaystyle \frac{d}{dp} \ ln[L(p)]$ = $\displaystyle \frac{x}{p} + \frac{1 - x}{1 - p}$

Is it correct or have i done some thing wrong now what should i do further...
My understanding from post #1 is that $\displaystyle L = f(0, p) \cdot f(1, p) = (1-p)p$. This has an obvious maximum.

However .... if the random sample is of size n then it's not clear to me what the n sample values of x are ....

6. ## Thanks for ur answer

Originally Posted by mr fantastic
My understanding from post #1 is that $\displaystyle L = f(0, p) \cdot f(1, p) = (1-p)p$. This has an obvious maximum.

However .... if the random sample is of size n then it's not clear to me what the n sample values of x are ....

Thanks Mr fantastic for the answer , but i need some steps so that i can understand how to solve such problems regarding Maximum likelihood .the steps which i have put in the prev post are the steps which i have sort from google and the books which i am using but dont know if the steps which i have put int he prev post are correct or no .

The question is a question asked i the prev years question paper which i am trying to solve by my self .

Need ur guidance and support in this . Thank you