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

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In summary, the conversation discusses a probability distribution in the form of an exponential distribution with a rate parameter of lambda/k. The significance of lambda and k in this notation is not clear, but it is noted that lambda is typically used as the rate parameter and has a specific meaning. The conversation also mentions the connection between this distribution and the beta function in Bayesian statistics.

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.

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?

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 $$\frac{\lambda}{k}$$.

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?)?