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

[thesandbox]

Homework Statement

X₁,X₂,...,X_k ~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

$\bar{X}$=n⋅p

$\bar{X}$² = n⋅p⋅(1-p) + n²⋅p²

$\bar{X}$² = $\bar{X}$⋅(1-p) + $\bar{X}$⋅$\bar{X}$

$\hat{p}$ = (($\bar{X}$)² + $\bar{X}$ - $\bar{X}$²) / $\bar{X}$

Thanks.

 

[Ray Vickson]

Homework Statement

X₁,X₂,...,X_k ~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

$\bar{X}$=n⋅p

$\bar{X}$² = n⋅p⋅(1-p) + n²⋅p²

$\bar{X}$² = $\bar{X}$⋅(1-p) + $\bar{X}$⋅$\bar{X}$

$\hat{p}$ = (($\bar{X}$)² + $\bar{X}$ - $\bar{X}$²) / $\bar{X}$

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 $$\bar{X} = \frac{\sum_{i=1}^k X_i}{k}?$$ If so, do you want $E (\bar{X}) \text{ and } \text{Var}( \bar{X})?$ If that is what you want to get, just use standard formulas for the mean and variance of $\bar{X}$ in terms of the means and variances of the $X_i .$ 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

 

[thesandbox]

MME := Method of Moments Estimator