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

  • A
  • Thread starter Thread starter Ad VanderVen
  • Start date Start date
  • Tags Tags
    Distribution Limit
Ad VanderVen
Messages
169
Reaction score
13
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.
 
Physics news on Phys.org
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
 
Last edited:
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?)?
 
Back
Top