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