Скачать или смотреть Sufficient Statistic for Variance in Normal with Known Mean | UPSC ISS 2024 Paper-2 | Problem-12

This video explains the derivation of the sufficient statistic for the parameter \(\theta\) (variance) in a normal distribution \(N(\mu, \theta)\) where the mean \(\mu\) is known. The likelihood function for a sample \(X_1, X_2, \ldots, X_n\) is expressed as:
\[
L(\theta) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi \theta}} \exp\left(-\frac{(X_i - \mu)^2}{2\theta}\right).
\]
The likelihood depends on the data only through the sum of squared deviations from the known mean:
\[
T = \sum_{i=1}^n (X_i - \mu)^2,
\]
which serves as the sufficient statistic for \(\theta\). This means the entire data's information about \(\theta\) is contained in this sum, simplifying inference and estimation. This concept is fundamental in statistics, especially for parameter estimation in normal models, and relevant for UPSC preparation.

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