Question

Which of following random variable is always non-negative?

• A. standard normal variate
• B. a chi-square variate
• C. a $t$-variate
• D. normal variate

Hint:

Any distribution derived directly from squared standard normal variables, such as Chi-square ($\chi^2$) and $F$-distributions, is always restricted to non-negative values.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The support of a random variable is the set of all possible values it can take.
A random variable is defined as non-negative if its support is restricted to $[0, \infty)$.

Step 2: Detailed Explanation:
Let us review the ranges of the given random variables:
- Standard Normal Variate ($Z$): Can take any real value. Range is $(-\infty, \infty)$.
- Normal Variate ($X$): Can take any real value. Range is $(-\infty, \infty)$.
- $t$-variate: Can take any real value. Range is $(-\infty, \infty)$.
- Chi-square variate ($\chi^2$): Formed by summing the squares of independent standard normal variates. Because of the squaring operation, it can never yield a negative value. Range is $[0, \infty)$.
Therefore, a chi-square variate is always non-negative.

Step 3: Final Answer
The correct option is (B).