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A comment on The Man Who Invented Modern Probability. Parent comment
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...
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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...