How does the Chi Square distribution emerge?

Question

BlackTom AI · Accepted Answer

To analyze how the Chi Square distribution emerges, we need to consider its foundational construction.
Option 1: Sum of ratios of sums of squared normals. This description does not align with the standard definition of a Chi Square variable. Chi Square arises from squared standard normals, not from ratios of sums of squared normals, so this claim misstates the fundamental mechanism.
Option 2: Sum of uniform random variables. Uniform variables are not squared nor arranged to produce the Chi Square distribution directly. The Chi Square distribution comes from normal variables, not uniforms, so this option is incorrect.
Option 3: Sum of squared standard normals. This matches the classic construction: if Z1, Z2, ..., Zk are independent standard normal random variables, then the statistic X = Z1^2 + Z2^2 + ... + Zk^2 follows a Chi Square distribution with k degrees of freedom. The intuition is that squaring standard normals captures their magnitude while removing sign, and summing across independent dimensions aggregates these squared deviations, yielding the Chi Square form.
In summary, only Option 3 correctly describes the emergence of the Chi Square distribution, while Options 1 and 2 mischaracterize the relevant operations or variables involved in its formation.

ERMCPS5350_006_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 8. Quiz for Class 8

How does the Chi Square distribution emerge?

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How does the Chi Square distribution emerge?

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