The Man Who Invented Modern Probability

Re:The Man Who Invented Modern Probability by dxbydt on Thursday August 22, 2013 @06:38PM (#6260547)

My, my, 2 mistakes on a silly uniform distribution! You must atone for this sin. Explain what the volume would be if the side came from a normal distribution over [1,5] (ie. mean 3, sigma 2/3) Otherwise I will personally call the MAA director on my speed dial & haul your ass back to stat101 :)

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.

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Re:The Man Who Invented Modern Probability by Cogito on Thursday August 22, 2013 @08:38PM (#6261022)

From [0]:

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.

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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...

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