# In comparing the exponential and Weibull distributions, why “rate” for exponential and “scale” for Weibull?

When comparing the exponential CDF with the Weibull CDF, we see:

Exponential: $${\displaystyle F(x)=1-e^{-\lambda x}}$$

Weibull: $${\displaystyle F(x)=1-e^{-(x/\lambda)^{k}}}$$

I understand how the exponential distribution models time to an event where occurrence intensity is a constant average (the $$\lambda$$, or rate parameter), while the Weibull distribution is similar, except that the probability increases or decreases over time (expressed via the $$k$$, or shape parameter).

In the literature I've read, most of the focus is on how $$k$$ is the real distinguisher in this regard, and indeed that when $$k=1$$, the Weibull distribution reduces to the exponential distribution. My question is more about the other parameter: why does exponential have a rate $$\lambda$$, while Weibull has a scale $$1/\lambda$$? What is the difference in meaning in this context between rate and scale? And why is scale expressed as an inverse of the actual parameter? I have even seen in the exponential distribution Wikipedia article that rate is sometimes called "inverse scale," but I am trying to get more of an intuition for the rate of the exponential distribution vs. the scale of the Weibull distribution beyond their being inverses of one another.

• Whether a rate, a scale, or something else is used for this parameter is purely a matter of how any particular writer chooses to express these distributions. – whuber Jul 29 '20 at 13:55