How to add variables with logistic distributions?

Question

i have X, Y, random logistic variables, how to add them taking into account the average and scale for each?

Logistics distribution . I ran the simulation in python, but I cannot get it to be exact.

I simulated getting a random number X, Y and saved the estimate for the value of X + Y. Then I did the same to get one random number with X + Y and checked another scale based on the original scale, but I can not fix the new scale to match

Answer 1

The sum of the two logistic random variables does not have a logistic distribution. However, the amount is approximately logical. You can justify this by asserting that the distribution of the logistics is approximately normal, and the sum of the two normal random variables is normal. ( This post explains how close the normal and logistic distributions are.)

Let's say that X1 has a value of m1, and the scale of s1 and X2 has an average value of m2 and a scale of s2. Then X1 + X2 has the value m1 + m2. X1 has a variance of pi ^ 2 s1 ^ 2/3, and X2 has a variance of pi ^ 2 s2 ^ 2/3, so X1 + X2 has a variance of pi ^ 2 (s1 ^ 2 + s2 ^ 2) / 3. That's for sure. We know the average and variance of the sum, although not its exact distribution. But if you agree to assume that the sum has an approximately logistic distribution, then the corresponding distribution of the logistics will have the value m1 + m2 and the scale sqrt (s1 ^ 2 + s2 ^ 2).

Answer 2

The numpy.random module (sorry for the strange link, it seems that it isn’t on the zero site now) has logistic , which should generate random numbers with logistic distribution correctly (did not check it personally, but I would be amazed if such a widely used package , like numpy , made incorrect statements). However, as mentioned in several comments, the sum of two random variables of the logistic distribution does not have the logistic distribution itself.