Which of the followings is NOT true about Chi-Square distribution?

Question

BlackTom AI · Accepted Answer

When evaluating statements about the Chi-Square distribution, it's useful to break down each claim and assess it against the defining properties.
Option 1: 'It is a continuous probability distribution.' This is true because the Chi-Square distribution is a continuous distribution over the nonnegative real numbers, with probability densities defined for x ≥ 0.
Option 2: 'It can assume any values from negative infinity to positive infinity.' This is false. The support of the Chi-Square distribution is [0, ∞); negative values are not possible since the distribution arises from sums of squared standard normals, which cannot be negative. This misstates the domain and is a common point of confusion with other distributions.
Option 3: 'The total area under the Chi-Square distribution curve is equal to 1.' This is true by the very definition of a probability density function: the integral over the entire support equals 1, which holds for the Chi-Square family as a valid distribution.
Option 4: 'The shape of the distribution depends on the parameter .' In the Chi-Square family, the shape is governed by the degrees of freedom parameter (often denoted k or ν). As ν changes, the skewness and peak shift accordingly, so this statement correctly describes the dependence of shape on the parameter.
In summary, the key issue among the options is the domain of the Chi-Square distribution. Everything else aligns with standard properties, while the second statement contradicts the fundamental support of the distribution.

STA2212.1HR1.JAN2025 Self Review Exercise 11

Which of the followings is NOT true about Chi-Square distribution?

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