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.

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    $\begingroup$ 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. $\endgroup$ – whuber Jul 29 at 13:55

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