关于Exp(θ,1)参数含义及Exp(θ,c)分布PDF的技术咨询

Hey there! Let's break this down step by step to clear up your confusion about the location-scale exponential distribution notation and its PDF.

First, let's ground this in what you already know: you're working with a location-shifted exponential distribution. The notation $ ext{Exp}( heta,1)$ uses two parameters:

• $ heta$: The location parameter (this shifts the distribution along the x-axis, hence why the support is $x > heta$ instead of $x > 0$ for the standard exponential).
• $1$: The scale parameter (this controls the spread of the distribution; when it's 1, there's no scaling applied beyond the location shift).

You noted the PDF is $f(x, heta) = e^{-(x- heta)}$ for $x > heta$. Let's connect this to the standard exponential distribution to see where the scale parameter 1 fits in:

• The standard exponential distribution (no location shift, scale=1) has PDF $f(x) = e^{-x}$ for $x > 0$.
• When we shift it by $ heta$, we replace $x$ with $x - heta$, giving $e^{-(x- heta)}$ for $x > heta$. Since the scale parameter is 1, we don't have an extra scaling factor in the PDF (like a denominator or multiplier)—that's why you don't see it explicitly altering the exponential term beyond the shift.

For the general case $ ext{Exp}( heta,c)$ where $c > 0$ is a positive constant (the scale parameter), we can derive the PDF using a location-scale transformation:

1. Start with a random variable $Y$ that follows a scale-only exponential distribution (no location shift): $Y \sim ext{Exp}(0,c)$. Its PDF is:
   $$f_Y(y) = \frac{1}{c}e^{-\frac{y}{c}} \quad ext{for } y > 0$$
2. To get the location-shifted version, define $X = Y + heta$. This shifts the entire distribution right by $ heta$, so the support becomes $x > heta$ (since $Y > 0$ implies $x - heta > 0$).
3. Using the change-of-variable formula for PDFs (the Jacobian determinant here is 1, since we're just adding a constant), the PDF of $X$ is:
   $$f_X(x) = f_Y(x - heta) = \frac{1}{c}e^{-\frac{(x - heta)}{c}} \quad ext{for } x > heta$$

To confirm this matches your original case: when $c=1$, this simplifies to $e^{-(x- heta)}$, exactly the PDF you provided. That's the key—your $ ext{Exp}( heta,1)$ is just the general location-scale exponential with a scale parameter of 1.

A quick note on notation: Different sources might use slightly different conventions for exponential distributions (some focus on rate parameters instead of scale), but in this $ ext{Exp}( heta,c)$ form, the first parameter is always the location shift, and the second is the scale factor.

内容的提问来源于stack exchange，提问作者Live Free or π Hard