Question

A random variable X is normally distributed with zero mean and unit variance. The variance of $X^2$ is

• A. 0.0
• B. 1.0
• C. 2.0
• D. 4.0

Hint:

Remember that the square of a standard normal variable is a $\chi^2$ variable with $1$ degree of freedom. The variance of any $\chi^2_k$ variable is always $2k$, which immediately gives $2 \times 1 = 2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem utilizes the properties of a standard normal random variable and its transformation into a chi-square distribution.
Key Formula or Approach:
If $X \sim N(0, 1)$, then $Y = X^2$ follows a chi-square distribution with 1 degree of freedom: \[ Y \sim \chi^2_1 \]

Step 2: Detailed Explanation:
The probability density function of a chi-square distribution with $k$ degrees of freedom has the following parameters: \[ \text{Mean} = k \] \[ \text{Variance} = 2k \]
Since $Y = X^2$ has $k = 1$ degree of freedom: \[ \text{Mean}(X^2) = 1 \] \[ \text{Variance}(X^2) = 2(1) = 2 \]
Alternatively, we can compute this using central moments of the standard normal distribution: \[ \text{Var}(X^2) = E[X^4] - (E[X^2])^2 \]
For $X \sim N(0, 1)$, the moments are: \[ E[X^2] = 1 \] \[ E[X^4] = 3\sigma^4 = 3(1)^2 = 3 \]
Substituting these moments: \[ \text{Var}(X^2) = 3 - (1)^2 = 2 \]
Thus, both methods yield a variance of 2.0.

Step 3: Final Answer:
The variance of $X^2$ is 2.0.