A The distribution that has a certain distribution as its limit case

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The discussion revolves around a specific probability distribution defined as f(t) = (λ e^(-λt/k))/k, which is identified as an exponential distribution with a rate parameter of λ/k. Participants explore whether this distribution is a limiting case of another distribution, similar to how the normal distribution serves as a limit for many others. Clarification is sought on the roles of parameters λ and k, particularly their significance in relation to the mean and how they affect the distribution's characteristics. The conversation also touches on the connection between this distribution and Bayesian statistics, questioning if the posterior distribution is being examined as a limit case. Ultimately, the focus remains on understanding the nature and implications of the exponential distribution defined in this context.
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I have a probability distribution of the following form:

$$\displaystyle f \left(t \right) \, = \, \frac{\lambda ~e^{-\frac{\lambda ~t }{k }}}{k }, \, 0 < t, \, 0< \lambda, \, k = 1, 2, 3, \dots$$

It seems that this distribution is a limiting case of another distribution. The question is what that other distribution might look like.
 
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f(t) is the density function for an exponential distribution. Other distribution?
 
Yes, I already knew that. Now I know that the normal distribution is the limiting case of many other distributions. Is there something similar to the exponential distribution?
 
Ad VanderVen said:
I have a probability distribution of the following form:

$$\displaystyle f \left(t \right) \, = \, \frac{\lambda ~e^{-\frac{\lambda ~t }{k }}}{k }, \, 0 < t, \, 0< \lambda, \, k = 1, 2, 3, \dots$$

Is that supposed to be a family of probability distributions? (one for each value of ##k##).

Or is that supposed to be a joint probability distribution for two variables ##(t,k)##?
 
No, it is simply an exponential distribution with rate parameter equal to $$\frac{\lambda}{k}$$.
 
Ad VanderVen said:
No, it is simply an exponential distribution with rate parameter equal to $$\frac{\lambda}{k}$$.
That's very confusing, we normally use ## \lambda ## as the rate parameter and it has a particular significance e.g. the mean is given by ## \mu = \frac 1 \lambda ##. What is the significance of ## \lambda ## and ## k ## in your notation e.g. what is the difference between ## (\lambda, k) = (1, 2) ## and ## (\lambda, k) = (2, 4) ##?

## \displaystyle {\lim_{n \to \infty} }n \operatorname{Beta} (1, n) ## is eqivalent to an exponential distribution with ## \lambda = 1 ## see https://en.wikipedia.org/wiki/Beta_function
 
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As stated ##\lambda/k## appears as such. They are not separate.
 
Just curious: Are you dealing with Priors/Posteriors in Bayesian Statistics( And looking at the Posterior as the limit?)?
 
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