How does the normal distribution emerge?

Pick between A , B and C

Question

How does the normal distribution emerge?

Pick between A , B and C

BlackTom AI · Accepted Answer

When thinking about how the normal distribution appears in probability theory, different routes lead to the same shape under the right limits. 
Option B: Sum of squared standard normals. Each squared normal variable follows a chi-square distribution with 1 degree of freedom, and the sum of several such squared normals yields a chi-square distribution, not a normal. So this path does not produce a normal distribution directly. 
Option C: Sum of ratios of sums of squared normals. This construction does not have a standard, widely recognized route to a normal distribution; ratios and sums of such terms typically lead to other distributions (e.g., F-distributions or more complex forms) rather than a simple normal. 
Option A: Sum of uniform random variables. Here, the central limit theorem comes into play: as you add many independent, identically distributed random variables with finite mean and variance (uniforms qualify), their sum (properly normalized) converges in distribution to a normal. This is a classic route to the normal curve, explaining how the normal distribution can emerge from basic, simple components. 
In summary, among the given options, the route most aligned with the emergence of a normal distribution via a fundamental limit theorem is the accumulation of many uniform random variables.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 11. Quiz for Class 11

How does the normal distribution emerge? Pick between A , B and C

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