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Can Two Independent Random Variables Sum to Uniform?

Audience:

Tags: calculusprobabilitylaplace-transformapplications

An unexpectedly long and at times crazy adventure spurred on by a simple question of probability. The realization that the sum of two independent uniform random variables on $[0, 1]$ is NOT a uniform random variable on $[0, 2]$ may initially be surprising. But then I asked: if you can’t sum uniform to uniform, what two independent random variables CAN you sum to it? We explore, and actually construct an answer, which winds through a lot of math territory, with whacky manipulations involving Laplace transforms, power series, and a fractional integral!

My intended audience is probably upper division undergraduate, definitely one who’s had a course in probability theory and differential equations. Possibly understandable by engineering students, as those delta functions and Laplace transform calculus very much were the bread and butter in a course I took called Signals and Systems.

Analytics

6.39 Overall score*

16 Rank

6 Votes

5 Comments

Comments

4.5

I think the topic is very interesting, but opening with that the sum of two independent random variables is uniform, was a big deturant to continue reading (and also destroys the motivation).

As someone who took Signals and Systems aswell, I could appricaite the beauty of the connections between convolution and the lapalce transform, however, I think outside of people who have that perspective, the explainer falls short.

4.2

After many pages you say, well, you can’t… How is it important to say you can’t and what does that mean practically? Also, has typo at the end. “The the number of different tools and areas of math that this touches upon, not to mention, having just a random factor of π showing up where there are no circles, is a real delight!”

6.3

Was a fun read! Overall a bit long, a bit too many walls of text. And basically no visualisations. (Which is in conflict with the header of the blog.)

7.7

I think the article does an overall good job to motivate the concept. But I would have personally liked to see more on that proof that it is impossible to sum two random variables to be uniform— especially given the title.

6.5

Clearly explained.