Question

Let X₁,X₂ ,...,Xₙ be a random sample from a Binomial distribution with parameters n and p, both unkown. Obtain estimators of n and p, using method of moments.

Answer 1

Since we know that

$$\text{E}[x]=np,$$

$$\text{Var}[x]=np(1-p),$$

we can express $p$ and $n$ so that they are

$$p=1-\frac{\text{Var}[x]}{\text{E}[x]},$$ $$n=\frac{\text{E}[x]}{p}=\frac{\text{E}^2[x]}{\text{E}[x]-\text{Var}[x]}.$$

On the other hand, we know that sample mean and sample variance are:

$$\text{E}[x]=x_\mu=\frac{1}{n}\sum_{i=1}^{n}x_i,$$

$$\text{Var}[x]=S(x)=\frac{1}{n}\sum^{n}_{i=1}(x_i-x_\mu)^2,$$

hence

$$p=1-\frac{\frac{1}{n}\sum^{n}_{i=1}(x_i-x_\mu)^2}{\frac{1}{n}\sum_{i=1}^{n}x_i},$$

$$n=\frac{(\frac{1}{n}\sum_{i=1}^{n}x_i)^2}{\frac{1}{n}\sum_{i=1}^{n}x_i-\frac{1}{n}\sum^{n}_{i=1}(x_i-x_\mu)^2}.$$