Question

Which of the following statements about chi-square distributions are true?

I. For all chi-square distributions, $P(x^2\ge 0)=1$

II. A chi-square distribution with fewer than $10$degrees of freedom is roughly symmetric.

III. The more degrees of freedom a chi-square distribution has, the larger the mean of the distribution.

a. I only

b. II only

c. III only

d. I and III

e. I, II, and III

Short answer

Option(d) is correct. Statement I and III are correct about chi-square distributions.

Step-by-step solution

Given information

We need to find correct statement about chi-square distributions.

Simplify

Statement I-

Because the chi-square distribution is specified solely for non-negative numbers, Statement I is correct.
Hence, $P(x^2\ge 0)=1$
Statement II-

It is incorrect since the chi-square distribution is always biassed to the right, hence it can't be a nearly symmetric distribution with less than $10$ degrees of freedom.
Statement III-

It is correct because the degrees of freedom are the mean of the chi-square distribution.
As a result, as the degrees of freedom increase, the mean rises.

Therefore, Statement I and III are correct.