# Answer to Question #86839 in Electrical Engineering for Anand

Question #86839
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.
1
2019-04-01T08:27:25-0400

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}."

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