Finding the MME for p of Bin(n,p)

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Homework Statement



X1,X2,...,Xk ~iid Bin(n,p) find the MME (Method of Moments Estimator) for p

Homework Equations



E[X] = n⋅p
Var[X] = n⋅p⋅(1-p)

Var(X) = E[X2] - [E[X]]2

The Attempt at a Solution



Does this look correct?

n⋅p⋅(1-p) = E[X2] - n2⋅p2

E[X2] = n⋅p⋅(1-p) + n2⋅p2

[itex]\bar{X}[/itex]=n⋅p

[itex]\bar{X}[/itex]2 = n⋅p⋅(1-p) + n2⋅p2

[itex]\bar{X}[/itex]2 = [itex]\bar{X}[/itex]⋅(1-p) + [itex]\bar{X}[/itex]⋅[itex]\bar{X}[/itex]

[itex]\hat{p}[/itex] = (([itex]\bar{X}[/itex])2 + [itex]\bar{X}[/itex] - [itex]\bar{X}[/itex]2) / [itex]\bar{X}[/itex]

Thanks.
 
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Answers and Replies

  • #2
Ray Vickson
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Homework Statement



X1,X2,...,Xk ~iid Bin(n,p) find the MME for p

Homework Equations



E[X] = n⋅p
Var[X] = n⋅p⋅(1-p)

Var(X) = E[X2] - [E[X]]2

The Attempt at a Solution



Does this look correct?

n⋅p⋅(1-p) = E[X2] - n2⋅p2

E[X2] = n⋅p⋅(1-p) + n2⋅p2

[itex]\bar{X}[/itex]=n⋅p

[itex]\bar{X}[/itex]2 = n⋅p⋅(1-p) + n2⋅p2

[itex]\bar{X}[/itex]2 = [itex]\bar{X}[/itex]⋅(1-p) + [itex]\bar{X}[/itex]⋅[itex]\bar{X}[/itex]

[itex]\hat{p}[/itex] = (([itex]\bar{X}[/itex])2 + [itex]\bar{X}[/itex] - [itex]\bar{X}[/itex]2) / [itex]\bar{X}[/itex]

Thanks.

I'm not sure what "MME" stands for; In have seen the terms MMSE, MLE, etc., but not MME. Anyway, do you want to take the sample mean [tex] \bar{X} = \frac{\sum_{i=1}^k X_i}{k}?[/tex] If so, do you want [itex] E (\bar{X}) \text{ and } \text{Var}( \bar{X})?[/itex] If that is what you want to get, just use standard formulas for the mean and variance of [itex] \bar{X}[/itex] in terms of the means and variances of the [itex] X_i . [/itex] You will get formulas very different from what you wrote (although I must admit I do not know exactly what you were trying to do).

RGV
 
  • #3
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MME := Method of Moments Estimator
 

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