关于概率密度函数$$P(x)=|x|e^{-x^2}$$是否有命名的技术问询

Hey there! Let's tackle your two questions one by one:

1. Do all probability density functions have a formal name? #

Nope, definitely not. Only PDFs that see widespread use across statistics, engineering, data science, or other quantitative fields get standardized names (think Normal, Uniform, Exponential, Gamma distributions, to name a few).

For many custom or niche PDFs—especially those that are minor variants of known distributions, or built for specific one-off use cases—you won’t find a widely recognized formal name. Instead, you’d typically describe them by their functional form or their relationship to more well-known distributions.

2. Does the PDF ( P(x) = |x|e^{-x2} ) have a dedicated name? #

Great question! First, let’s confirm it’s a valid PDF: integrating over all real numbers gives ( \int_{-\infty}^{\infty} |x|e^{-x2} dx = 1 ), which checks out.

Now, for the naming part: this distribution doesn’t have a single, universally accepted standard name that’s as common as, say, the Normal distribution. That said, it’s closely related to several named distributions, and you might see it referred to in a few ways:

• Symmetrized Rayleigh Distribution: If you take a Rayleigh-distributed random variable (which is non-negative) and flip its sign with 50% probability to make it symmetric around 0, you get this PDF. The Rayleigh distribution’s PDF for non-negative ( x ) is ( \frac{x}{\sigma^2}e{-x^2/(2\sigma2)} ); setting ( \sigma^2 = 1/2 ) gives ( 2x e^{-x2} ) for ( x \geq 0 ), and symmetrizing this (adding the mirrored negative half) results in ( |x|e^{-x2} ).
• Special Case of the Symmetrized Half-Gamma Distribution: The Half-Gamma distribution describes non-negative random variables; symmetrizing it (extending to negative values with mirrored density) gives a family of symmetric distributions, and your PDF is a member of this family with specific shape/scale parameters.
• Symmetric Generalized Gamma Distribution: It fits the form of a symmetric generalized Gamma distribution with parameters ( \mu=0 ), ( \alpha=1 ), ( \beta=2 ) (the generalized Gamma distribution’s symmetric extension includes absolute-value terms for even ( \beta )).

In most contexts, unless you’re working in a specialized field where this exact distribution has a local convention, it’s perfectly acceptable to just describe it by its functional form or reference its relationship to Rayleigh/Half-Gamma distributions.

内容的提问来源于stack exchange，提问作者thedude