Variance Calculation for Y = Z^2

We are given a normal random variable $X$ with mean $\mu$ and variance $\sigma^2$, denoted as $X \sim N(\mu, \sigma^2)$.

A standardized variable $Z$ is defined as $Z = \frac{X - \mu}{\sigma}$. This transformation makes $Z$ a standard normal random variable, meaning $Z \sim N(0, 1)$.

We need to find the variance of $Y = Z^2$.

Properties of Standard Normal Variable Z

• For $Z \sim N(0, 1)$, the expected value is $E[Z] = 0$.
• The variance is $Var(Z) = E[Z^2] - (E[Z])^2 = 1$.
• Since $E[Z] = 0$, we have $E[Z^2] - 0^2 = 1$, which means $E[Z^2] = 1$.

Relationship between Y and Chi-Squared Distribution

The variable $Y$ is defined as $Y = Z^2$. Since $Z \sim N(0, 1)$, the square of $Z$, which is $Z^2$, follows a Chi-squared distribution with 1 degree of freedom. This is denoted as $Y \sim \chi^2(1)$.

Calculating Variance of Y

The variance of a Chi-squared distribution with $k$ degrees of freedom ($\chi^2(k)$) is given by the formula $Var = 2k$.

In our case, $Y \sim \chi^2(1)$, so $k=1$. Applying the formula:

$ Var(Y) = 2k = 2 \times 1 = 2 $

Therefore, the variance of $Y$ is 2.