Solution Found!
If Y has a binomial distribution with parameters n and p,
Chapter 8, Problem 17E(choose chapter or problem)
If Y has a binomial distribution with parameters n and p, then \(\hat{p}_{1}=\mathrm{Y} / \mathrm{n}\) is an unbiased
estimator of p. Another estimator of p is \(\hat{p}_{2}=(Y+1) /(n+2)\)
Derive the bias of \(\hat{p}_{2}\)Derive \(\operatorname{MSE}\left(\hat{p}_{1}\right) \text { and } \operatorname{MSE}\left(\hat{p}_{2}\right)\)For what values of p is \(\operatorname{MSE}\left(\hat{p}_{1}\right)<\operatorname{MSE}\left(\hat{p}_{2}\right)\)?
Equation Transcription:
and
Text Transcription:
\hatp_1=Y / n
\hatp_2=(Y+1) /(n+2)
\hatp_2
MSE hat p_1 and \MSE(hatp_2)
MSE hat p_1 right < MSE hat p_2
Questions & Answers
QUESTION:
If Y has a binomial distribution with parameters n and p, then \(\hat{p}_{1}=\mathrm{Y} / \mathrm{n}\) is an unbiased
estimator of p. Another estimator of p is \(\hat{p}_{2}=(Y+1) /(n+2)\)
Derive the bias of \(\hat{p}_{2}\)Derive \(\operatorname{MSE}\left(\hat{p}_{1}\right) \text { and } \operatorname{MSE}\left(\hat{p}_{2}\right)\)For what values of p is \(\operatorname{MSE}\left(\hat{p}_{1}\right)<\operatorname{MSE}\left(\hat{p}_{2}\right)\)?
Equation Transcription:
and
Text Transcription:
\hatp_1=Y / n
\hatp_2=(Y+1) /(n+2)
\hatp_2
MSE hat p_1 and \MSE(hatp_2)
MSE hat p_1 right < MSE hat p_2
ANSWER:Step 1 of 6
The variable Y follows the binomial distribution with parameters n and p.
Mathematically,
The expected value and the variance of Y is given as,
Let,
