Which of the following is true about chi-square distribution?

Question

BlackTom AI · Accepted Answer

Chi-square distribution is a family of distributions indexed by its degrees of freedom (k), so the statement that “It is characterized by degrees of freedom” is correct.

1. Relevant knowledge points and concepts
- Definition: If Z1,...,Zk are independent standard normal random variables N(0,1), then X = Σ_{i=1}^k Zi^2 has a chi-square distribution with k degrees of freedom, written χ^2_k.
- Parameters: The single parameter k (degrees of freedom) determines the shape, mean, variance, and skewness.
- Moments: E[X] = k and Var(X) = 2k. Support is [0, ∞) and the distribution is right-skewed for small k, approaching normal as k grows.

2. Step-by-step reasoning
- Start from the definitional sum of squares: X = Σ Zi^2.
- Use linearity of expectation: E[X] = Σ E[Zi^2] = k·E[Z^2]. For Z ~ N(0,1), E[Z^2] = Var(Z) = 1, so E[X] = k.
- For variance, note Var(Z^2) = E[Z^4] − (E[Z^2])^2. For standard normal, E[Z^4] = 3, so Var(Z^2) = 3 − 1 = 2. Independence gives Var[X] = Σ Var(Zi^2) = k·2 = 2k.
- These formulas explicitly show the distribution’s properties depend on k (degrees of freedom).

3. Intermediate calculations (example)
- If k = 5: E[X] = 5; Var(X) = 2·5 = 10. This illustrates how the moments scale with degrees of freedom.

4. Why the correct answer is correct
- Because the chi-square distribution is fully specified by its degrees-of-freedom parameter k (it is not a single fixed curve but a family indexed by k), the statement “It is characterized by degrees of freedom” is true.

5. Why the other options are incorrect
- “It is computed using the Excel function CHISQ.INV.TEST.”: This names a specific software function; chi-square is a theoretical distribution, not “computed” by one Excel routine. Excel provides various functions (CHISQ.DIST, CHISQ.DIST.RT, CHISQ.INV) but that option misstates the nature of the distribution.
- “It is a special distribution and is denoted by (C^2).”: The conventional notation is χ^2 (chi-squared), not “C^2.” The phrasing is therefore incorrect.
- “It has a left-tail area equal to the expected value of the sample.”: Tail areas are probabilities and are not generally equal to the expected value; this statement is nonsensical for chi-square.

6. Conclusion
The correct choice is “It is characterized by degrees of freedom,” which matches the provided answer.

Which of the following is true about chi-square distribution?单项选择题

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