IGNOU MST 21 2026 - Suppose the number of breakdowns per month in a factory follows a Pois

Question

Suppose the number of breakdowns per month in a factory follows a Poisson distribution with parameter λ.

(i) Find the Cramer-Rao lower bound for an unbiased estimator of λ.

(ii) Obtain the UMVUE of λ.

(i) Find the Cramer-Rao lower bound for the variance. (ii) Also, find the UMVUE of $\lambda$.

18 Mar 2026

Answer :

Word Count : 358

Let (X_1, X_2, \dots, X_n) be a random sample from a Poisson distribution with parameter (\lambda), representing the number of breakdowns per month. The probability mass function is given by (P(X_i = x_i) = \frac{e^{-\lambda} \lambda^{x_i}}{x_i!}), where (x_i = 0,1,2,\dots). The likelihood function for the sample is [ L(\lambda) = \prod_{i=1}^n \frac{e^{-\lambda} \lambda^{x_i}}{x_i!} = e^{-n\lambda} \lambda^{\sum x_i} \prod_{i=1}^n \frac{1}{x_i!}. ] Taking logarithms, the log-likelihood function is [ \ell(\lambda) = -n\lambda + \left(\sum_{i=1}^n x_i\right)\log \lambda - \sum_{i=1}^n \log(x_i!). ] To find the Cramér-Rao lower bound, we first compute the Fisher information. The first derivative of the log-likelihood is [ \frac{d\ell}{d\lambda} = -n + \frac{\sum x_i}{\lambda}. ] The second derivative is [ \frac{d^2\ell}{d\lambda^2} = -\frac{\sum x_i}{\lambda^2}. ] The Fisher information is given by [ I(\lambda) = -E\left[\frac{d^2\ell}{d\lambda^2}\right] = -E\left[-\frac{\sum x_i}{\lambda^2}\right] =

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