Question

Part 3 7. A detector counts the number of particles emitted from a radioactive source over the course of 10-second intervals, For 180 such 10-second intervals, the folowig counts were observed: Count # intervals 23 34 26 13 This table states, for example, that in 34 of the 10-second intervals a count of 2 was recorded. Sometimes, however, the detector did not function properly and recorded counts over intervals f length 20 seconds. This happened 20 times and the recorded counts are Count # intervals 9 Assume a Poisson process model for the particle emission process. Let λ > 0 (time unit 1 sec.) be the unknown rate of the Poisson process. (a) Formulate an appropriate likelihood function for the described scenario and derive the maximum likelihood estimator λ of the rate λ Compute λ for the above data (b) What approximation to the distribution of A does the central limit theorem suggest? (Note that the sum of all 200 counts has a Poisson distribution. What is its parameter?)

PLEASE ONLY ANSWER PART B - THE SOLUTION TO A SHOULD BE THE WEIGHTED AVERAGE OF EVENTS, THAT IS XBAR

Answer 1

$(b)~The~likelihood~function~of~\lambda=L(\lambda)=\frac{e^{-n\lambda}\lambda^{\sum_{i=1}^nx_i}}{\prod_{i=1}^nx_i!}\\ \log L(\lambda)=l(\lambda)=-n\lambda&plus;\sum_{i=1}^nx_i\log \lambda&plus;\sum_{i=1}^n\log x_i\\ \frac{\mathrm{d} l(\lambda)}{\mathrm{d} \lambda}=-n&plus;\frac{1}{\lambda}\sum_{i=1}^nx_i=0~or,~\hat{\lambda}=\frac{1}{n}\sum_{i=1}^nx_i=\bar{x}=MLE~of~\lambda\\ since~\frac{\mathrm{d}^2 l(\lambda)}{\mathrm{d} \lambda^2}=-\frac{1}{\lambda^2}\sum_{i=1}^nx_i<0~for~\lambda=\hat{\lambda}.\\ -E\left( \frac{\mathrm{d}^2 l(\lambda)}{\mathrm{d} \lambda^2}\right )=\frac{1}{\lambda^2}\times n\lambda=\frac{n}{\lambda}\\ Then~from~central~limit~theorem,~\sqrt{n}(\hat{\lambda}-\lambda)\overset{L}{\rightarrow} N\left(0, \frac{1}{-E\left( \frac{\mathrm{d}^2 l(\lambda)}{\mathrm{d} \lambda^2}\right )}=\frac{\lambda}{n}\right ).\\ Hence~\hat{\lambda}\overset{L}{\rightarrow}N\left(\lambda,~\frac{\lambda}{n} \right )~and~its~parameter=\lambda~since~n~is~given.$