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What does $X^2 \sim \sigma^2 \chi_1^2$ mean? I’m looking at this:

Mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$?

and trying to make sense of it. In particular, I’m not following how the expectation and variance results follow.

James Bradley
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2 Answers2

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It means that $X^2$ is distributed as a scaled chi-squared distribution with one degree of freedom, with scale parameter $\sigma^2$. That is to say,

$$\chi_1^2 \sim \operatorname{ChiSquare}(1), \\ f_{\chi^2_1}(t) = \frac{e^{-t/2}}{\sqrt{2\pi t}}, \quad t > 0,$$

and so $X^2 = \sigma^2 \chi_1^2$ has density

$$f_{X^2}(x) = \frac{e^{-x/(2\sigma^2)}}{\sqrt{2\pi x} \sigma}, \quad x > 0.$$

That is all that is meant. In fact, I would argue that the expression should read $$X^2 = \sigma^2 \chi_1^2$$ and not $$X^2 \sim \sigma^2 \chi_1^2,$$ because $\sim$ means "distributed as," and it is sloppy and imprecise to say that something is distributed as $\sigma^2$ times a distribution. Rather, the random variable $X^2$ is algebraically equivalent to $\sigma^2 \chi_1^2$ where $\chi_1^2$ is a random variable with chi-squared distribution.

heropup
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  • I more often see $\chi^2_k$ denote the distribution (i.e. similar to how $\mathcal{N}(\mu, \sigma^2)$ is used, like what you denote by $\text{ChiSquare}(1)$) rather than a random variable having that distribution. For example, see the Wikipedia page, where it is almost always to the right of the $\sim$ symbol. – angryavian Mar 29 '22 at 02:50
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$X^2 \sim \sigma^2 \chi^2_1$ is an abuse of notation for "$X^2/\sigma^2 \sim \chi^2_1$" or more explicitly "$X^2 / \sigma^2$ follows a chi-squared distribution with $1$ degree of freedom."

angryavian
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