How does the Chi Square distribution emerge?

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BlackTom AI · Accepted Answer

Question restatement: How does the Chi Square distribution emerge?
Option provided: Sum of squared standard normal variables
Now let's analyze the concept step by step.
- Core idea: The chi-square distribution with k degrees of freedom arises as the distribution of the sum of the squares of k independent standard normal variables. If Z1, Z2, ..., Zk are iid N(0,1), then X = Z1^2 + Z2^2 + ... + Zk^2 follows a χ²_k distribution. This is a fundamental result in probability theory and underpins many statistical tests, including goodness-of-fit and tests of independence.
- Why this option is correct: It matches the canonical construction of the chi-square distribution: a sum of squared standard normals. The degrees of freedom correspond to the number of squared terms in the sum. The chi-square distribution is defined precisely by this sum, and properties like its mean being k and variance being 2k follow from this representation.
- Why this option is not claiming false information: There are alternative ways to characterize the chi-square distribution (e.g., as a gamma distribution with shape k/2 and scale 2, or as the distribution of the sum of squared independent normals), but all these views are equivalent. The essential mechanism is the sum of squares of independent standard normal variables.
- Any potential confusion to avoid: If one thinks of chi-square as simply a squared normal variable, that would be incorrect; it is specifically the sum of squares of multiple independent standard normals. Also, chi-square is not the distribution of a single normal variable or a linear combination of normals without squaring.
- Summary takeaway: The defining emergence of the chi-square distribution is exactly the sum of the squares of k independent standard normal variables, which aligns with the standard definition and most common applications in hypothesis testing and inference.

ERMCPS5350_005_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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