What is the name of the parameter that determines the shape of the chi-square distribution?

Question

BlackTom AI · Accepted Answer

The chi-square distribution is parameterized by the degrees of freedom (commonly denoted k), so that k determines its shape.

1. Relevant knowledge points and concepts
- A chi-square random variable with k degrees of freedom is defined as the sum of the squares of k independent standard normal variables: X = Σ_{i=1}^k Z_i^2.
- Its probability density function depends explicitly on k: f(x; k) = (1 / (2^{k/2} Γ(k/2))) x^{k/2 - 1} e^{-x/2} for x > 0.
- Key moments: E[X] = k and Var(X) = 2k, showing mean and variance are functions of k.

2. Step-by-step reasoning
- Start with k independent Z_i ~ N(0,1). Squaring and summing yields X = Σ Z_i^2.
- The number k of squared normals is what changes the distribution’s shape: more terms shift mass and change skewness.
- The PDF above includes k in the normalization constant, the exponent of x, and the gamma function; thus any change in k alters the density across x.

3. Intermediate calculations shown
- PDF: f(x; k) = (1 / (2^{k/2} Γ(k/2))) x^{k/2 - 1} e^{-x/2}.
- Mean: E[X] = k (derivable from moment properties of the gamma distribution).
- Variance: Var(X) = 2k.

4. Why the correct answer is correct
- Because the chi-square distribution is defined directly by summing k squared standard normals, the integer k (degrees of freedom) is the fundamental parameter controlling the distribution’s shape, location of mass, skewness, and moments.

5. Why other options are incorrect
- Proportion: unrelated to the family; chi-square is continuous and not parameterized by a proportion.
- Variance: while the distribution’s variance equals 2k, variance is a derived moment (function of k), not the defining parameter.
- Mean: similarly, mean = k is a moment determined by k, not the parameter that defines the family.

6. Conclusion
The parameter that determines the shape of the chi-square distribution is Degrees of freedom, matching the provided answer.

What is the name of the parameter that determines the shape of the chi-square distribution?Single choice

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