Central Limit Theorem can be formulated in the following way: "Let Sn be the sum of n independent, identically distributed random variables X1, X2, ..., Xn, then, as
𝑛
⟶
∞
, Sn converges to ________"

Fill in the blank:

Question

Central Limit Theorem can be formulated in the following way: "Let Sn be the sum of n independent, identically distributed random variables X1, X2, ..., Xn, then, as 
𝑛
⟶
∞
, Sn converges to ________"

Fill in the blank:

BlackTom AI · Accepted Answer

The question asks for the limiting distribution of the sum S_n of n independent, identically distributed random variables X1, X2, ..., Xn as n grows without bound.

Under the Central Limit Theorem, when X1, X2, ..., Xn are i.i.d. with finite mean μ and finite finite positive variance σ^2, the sum S_n = X1 + X2 + ... + Xn, properly centered and scaled, converges in distribution to a normal distribution. In the classical formulation without standardization, S_n itself does not converge to a fixed distribution, but the standardized sum (S_n − nμ) / (σ√n) converges in distribution to a standard normal N(0,1). If we interpret the question’s blank in the common phrasing that the sum “converges to” a normal distribution in the sense of its standardized form, the intended completion is: the Normal distribution.

Given the answer provided is Normal distribution, here is how this fits with the CLT concepts:
- The CLT asserts that the sum, when centered by its mean nμ and scaled by the standard deviation σ√n, becomes approximately normal for large n.
- Therefore, in the limit, the distribution of the standardized sum tends to Normal(0,1), and the unstandardized sum is described by a normal law after appropriate normalization.

Since the fill-in-the-blank asks for the limiting distribution related to Sn, the canonical, widely-cited completion is the Normal distribution.

Note: If the question intended the exact phrasing “converges to a Normal distribution” without standardization, that is an informal but common shorthand for the CLT result. The rigorous statement uses convergence in distribution of (S_n − nμ)/(σ√n) to N(0,1).

ERMCPS5350_005_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 7. Quiz for Class 7

Central Limit Theorem can be formulated in the following way: "Let Sn be the sum of n independent, identically distributed random variables X1, X2, ..., Xn, then, as 𝑛 ⟶ ∞ , Sn converges to ________" Fill in the blank:

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