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Scootertaj

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**1. Suppose X~B(5,p) and Y~(7,p) independent of X. Sampling once from each population gives x=3,y=5. What is the best (minimum-variance unbiased) estimate of p?**

## Homework Equations

[tex]P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}[/tex]

## The Attempt at a Solution

My idea is that Maximum Likelihood estimators are unbiased, and have asymptotic variance = Cramer-Rao lower bound. Also, C-R lower bound = minimum variance of unbiased estimators.

So, since X and Y independent, [tex]X+Y \sim B(5+7,p)=B(12,p)[/tex]

Thus, can we just compute the likelihood function and take the derivative?

[tex]L = \binom{12}{8}p^8(1-p)^4[/tex]

[tex]\frac{dL}{dp} = 8p^7(1-p)^4 - 4p^8(1-p)^3[/tex]

Thus, [tex]p=8/12=2/3[/tex]

Is that legit?

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