# Calculate $E[X]$ using polynomial approximation of CDF

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.

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It depends on the behavior of F(t). Do keep in mind that polynomials don't have asymptotes... – J. M. Nov 18 '11 at 9:32
Structure of CDF is usually more close to sigmoid 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? – Ilya Nov 18 '11 at 9:33
Presumably $F(0) = 0$ 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. – Dilip Sarwate Nov 18 '11 at 16:42

$$\int_{-\infty}^\infty xp(x)\mathrm dx=\lim_{L\to\infty}\left(\left[xF(x)\right]_{-L}^L-\int_{-L}^L F(x)\mathrm dx\right)\;,$$
and you want to approximate that last integral, given the ability to sample values of $F$. You can apply any quadrature methods, e.g. Gaussian quadrature, that seem suitable for the problem; this will implicity approximate $F$ by polynomials, but in a particularly efficient way. You may be able to use any knowledge about the structure of $F$ that you have in choosing the quadrature method or perhaps transforming the integral before evaluating it.
@Dilip: What does that mean? $\infty-\infty$ isn't defined, whereas the expression in parentheses and its limit for $L\to\infty$ are well-defined. If you mean that the individual terms tend to infinity, that's true, but that's not a problem. It might be a problem numerically if you choose $L$ too big and then get cancellation, but since the OP already mentioned integrating over a "suitable range" in the question, I assume that $p$ has approximately bounded support and one can choose $L$ so as to avoid such cancellation problems. – joriki Nov 18 '11 at 10:08
@Dilip: Sorry, I still don't understand what you're saying. I agree with everything in your first sentence, I just don't understand why you think this is a problem or why you described this as "tending to $\infty-\infty$". As far as I can tell, your suggestion is equivalent to mine if you replace $-L$ by $0$. – joriki Nov 18 '11 at 13:44
@Dilip: I'm not sure I do :-). I was genuinely trying to understand what you were saying. Your suggestion using $1-F(x)$ seemed elegant, and I'd like to incorporate it into the answer; my only problem was that I don't understand why you think there are principal issues with things tending to infinity in my formulation but not in that formulation. – joriki Nov 18 '11 at 14:25
For a positive random variable $X$, $$E[X]=\int_0^\infty [1-F(x)]\mathrm dx$$ is a standard result that has been discussed many times on math.SE as well (see, e.g. here) and so you can incorporate it into your answer without necessarily referencing me. In this formulation, writing this integral for $E[X]$ as a difference of integrals and then computing each separately numerically (as a neophyte (I don't mean you!) might try to do) is a cause for concern. – Dilip Sarwate Nov 18 '11 at 14:43