aside, I recently interviewed a candidate for a "Big Data" position. He got quite chatty about Kolmogorov & Measure Theory, so I quickly cooked up an interesting problem to relax his nerves. I opened the interview with "If the side of a cube comes from a uniform distribution between 1 and 5 inches, what's the expected volume of the cube ?" I thought he'll give me the answer in 10 seconds & we'll move on but instead he was completely stumped & just stared blankly. I asked him to use the whiteboard but he simply drew a cube and wrote a^3, a~[1,5], and then went back to staring at it. Finally after 10 minutes of pin-drop silence, my partner, a non-math guy, took over the interview & asked him some Hadoop related technology questions & he started speaking again.
In a bit more detail: take the integral of x^3 (the volume function) from 1 to 5 and you get 5^4/4 - 1^4/4 = 624/4 = 156. Then we divide by the length of the line segment (4) to get 39.
If the side lengths have to be integers, then it's just (1 + 8 + 27 + 64 + 125)/5 = 45 inches.
Edit: corrected division mistake, thanks to dxbydt
Edit 2: and corrected a second mistake thanks to tzs.
Well, one of the mistakes:
> If the side lengths have to be integers, then it's just (1 + 8 + 27 + 125)/4 = 40.25 inches
If I'm not mistaken, 4 is an integer, and so should not have cruelly snubbed when you sent out invitations to your summation of the cubes of the integers from 1 to 5. Sure, 4 is not a cool integer like the others in 1 to 5...he's got a reputation as a square, but that doesn't mean he doesn't have feelings.
But seriously, you write rather well on Quora & I'm sure the crowd would want to know what happens to the volume were you to choose a normal distribution instead of a uniform, so I hope you respond.
Let X be a random variable and g be any function.
If X is continuous, then the expectation of g(X) is defined as,
E[g(X)] = \int_−∞^∞ g(x)f(x) dx,
where f is the probability density function of X.
---
For us, with mean 3 and sigma 2/3, f = 3/2 e^(-9/8 (x-3)^2)/sqrt(2 π)
We need to integrate numerically; using Wolfram [1] we get an estimate of 30.8099.
If we ask for the expected value over the entire distribution we get 31.
[0] http://imai.princeton.edu/teaching/files/Expectation.pdf
[1] http://www.wolframalpha.com/input/?i=integrate+3%2F2+e%5E%28...
[2] http://www.wolframalpha.com/input/?i=expected+value+of+x%5E3...
1/3 of the cubes have volume 1, 1/3 have volume 27, and 1/3 have volume 125. The expected value of the volume is 51. Think about why it is 51, not 27, and you'll see why it is not 27 for the continuous case.
In general, if you have a random variable X, and some function of that random variable, f(X), then it is NOT true that expected value of f(X) = f(expected value of X).
from random import random
sum([(random()*4+1)**3 for x in xrange(1000000)])/1000000
=> ~39
sum(random.uniform(1, 5)**3 for i in range(10000)) / 10000
The historical anecdote my thesis advisor was telling about origins of his (i mean founded by him) branch of the science was along the lines "Americans had computers and could get good numerical solutions, while we didn't have computers and thus were forced to work on analytical solution". Well, that analytical solution he produced happened to be a great insight into the problem space and still continues to be the key tool for the research in that area almost 60 years later.
What does it mean in this context? Non-linear probability?
And this is what "big data" and NLP were like before computers :).
One can only imagine what he could have done with a modern computer. (Which is particular hard to imagine because he's responsible for advances which ultimately led to these same modern computers...)
Also, as an aside, I'm very impressed by the quality of nautil.us articles. I've come a across a few randomly, and they've been uniformly high in quality, interesting and insightful.
Moreover, unlike much popular science/math reporting, the articles are not ridden with obvious errors and yet are easily followed by somebody not in the field. An impressive combination. (Not surprising for this particular article, I suppose: it was written by a professor teaching the history of mathematics at MIT.)