Strong Law of Large Numbers | UPSC ISS 2022 Paper-1 | Problem-41 | RitwikMath
Автор: RitwikMath
Загружено: 2026-01-15
Просмотров: 13
Описание:
This video explains a fundamental result from *probability theory**—the **Strong Law of Large Numbers (SLLN)**—which is frequently tested in **UPSC Civil Services (Statistics Optional)* and **ISS examinations**.
Let \(X_1, X_2, \ldots\) be a sequence of **independent and identically distributed (i.i.d.) random variables**.
🔹 If:
\[
E(X_k) = \mu \quad \text{and} \quad \operatorname{Var}(X_k) \infty
\]
then, by the **Strong Law of Large Numbers**:
\[
\frac{1}{n}\sum_{k=1}^n X_k \xrightarrow{\text{a.s.}} \mu
\]
That is, the sample mean converges to the population mean **almost surely**.
✅ Hence, **Option A is the correct answer**.
This result is a cornerstone of **probability theory**, forming the theoretical basis of **statistical estimation**, and is essential for **Statistics Optional**, **ISS**, and advanced probability exams.
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