Скачать или смотреть Part-2: CRLB and Sample Mean Variance: Normal vs Bernoulli | UPSC ISS 2024 Paper-2 | Problem-14

This video explains how the variance of the sample mean relates to the Cramér-Rao lower bound (CRLB) in two important settings:

1. *Normal Distribution:* For independent and identically distributed samples from \(N(\theta, \sigma^2)\) with unknown \(\theta\), the variance of the sample mean \(\bar{X}\) perfectly attains the CRLB, which is \(\frac{\sigma^2}{n}\). The Fisher information for one observation is derived and generalized to the sample, demonstrating that the sample mean is an efficient estimator in the normal case.

2. *Bernoulli Distribution:* For Bernoulli trials with success probability \(\theta\), the video introduces the problem, emphasizing that the variance of the sample mean behaves differently compared to the normal case.

This contrast provides insights into when sample means are efficient estimators and the nuances of applying CRLB across distributions, essential for mastering statistical inference in UPSC exam preparations.

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