I have a black box called $F(t)$ ($~$($P~(X\le t)~$, $X$ is random variable) with me where I don't have any information on the exact expression of $F(t)$. But if I supply a $t\ge 0$ I will get a value of $F(t)$ from the black box as output. I want to calculate $E[X]$. I want to fit $F(t)$ values against a polynomial of the form $\sum a_{i}t^{i}$ and then integrate over a suitable range to get the expectation. Is this a good approach to calculate $E[X]$? Also, how should I choose $t$ values. Suppose I want to choose hundred $t$ values. Should I choose them as equally-spaced.
Calculate $E[X]$ using polynomial approximation of CDF
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probability
polynomials
approximation
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0It depends on the behavior of F(t). Do keep in mind that polynomials don't have asymptotes... – 2011-11-18
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0Structure of CDF is usually more close to [sigmoid](http://en.wikipedia.org/wiki/Sigmoid_function) than to polynomials so I wouldn't say it's the best idea. Anyway, if you're working with black-box, why not to approximate $\mathsf E[X]$ directly? – 2011-11-18
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0Presumably $F(0) = 0$ because if not, getting a good estimate of $E[X]$ might be difficult. Also, you say "Suppose I want to choose hundred $t$ values. Should I choose them as equally-spaced?" You might want to use an _adaptive_ strategy. No point in deciding on the $100$ values ahead of time and asking for, say, $F(0)$, $F(1), \ldots, F(99)$ and getting response $0$ in all cases because $X$ takes on values only in $[200, 800]$, e.g. a GRE score. So, use the first few calls to the black box to learn a little about $F(t)$, and then decide on the strategy. – 2011-11-18