The distribution that has a certain distribution as its limit case

[Ad VanderVen]

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.

 

[mathman]

f(t) is the density function for an exponential distribution. Other distribution?

 

[Ad VanderVen]

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?

 

[Stephen Tashi]

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 $(t,k)$?

 

[Ad VanderVen]

No, it is simply an exponential distribution with rate parameter equal to $$\frac{\lambda}{k}$$.

 

[pbuk]

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

 

[mathman]

As stated $\lambda/k$ appears as such. They are not separate.

 

[WWGD]

Just curious: Are you dealing with Priors/Posteriors in Bayesian Statistics( And looking at the Posterior as the limit?)?