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I was solving for the upper bound of a complicated probability distribution and I was able to use Mathematica to get a decimal approximation. I dropped that number into WolframAlpha to see if perhaps there was a closed-form solution.

enter image description here

This is one of the possibilities, alas I have no idea what script B refers to... I am not sure if this is even related to by probability density, at this point I want to know what it is just out of curiosity.

The pdf I was working on was (starting at x=1, ending at the aforementioned decimal):

$$f_X(x)=\frac{11}{10} \frac{e^x-1}{x^2}$$

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  • $\begingroup$ P.S. ExpIntegralEi[ ] was coming into play heavily when solving this. $\endgroup$ Jan 6 '18 at 11:52
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    $\begingroup$ I'm voting to close this question as off-topic because it is solely about Wolfram|Alpha, and does not concern Mathematica. $\endgroup$
    – Szabolcs
    Jan 6 '18 at 13:18
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    $\begingroup$ I only used Alpha at the end to try to find a closed-form solution for my numerical approximation that I got in Mathematica. Usually there is some cross-over as Wolfram Alpha basically runs on top of Mathematica... I guess I could rephrase as to is Script B a function of some sort within Mathematica? $\endgroup$ Jan 6 '18 at 13:23
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    $\begingroup$ I think I found it mathworld.wolfram.com/NortonsConstant.html $\endgroup$ Jan 6 '18 at 13:27
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    $\begingroup$ The bound does not necessarily need to have a closed form. $\endgroup$
    – ilian
    Jan 6 '18 at 17:22
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The CDF of the PDF is

int = Assuming[ub > 1, Integrate[11/10 (E^x - 1)/x^2, {x, 1, ub}]]

-((1/(10*ub))*(11*(-1 + E^ub + ub - E*ub + ub*ExpIntegralEi[1] - 
            ub*ExpIntegralEi[ub])))

To be a valid distribution the total probability must be 1

xmax = ub /. FindRoot[int == 1, {ub, 2}, WorkingPrecision -> 20]

(* 1.5684317087366619977 *)

int /. ub -> xmax

(* 1.000000000000000000 *)

Getting Norton's constant from WolframAlpha

ℬ = WolframAlpha["Norton's constant", {{"DecimalApproximation", 1}, 
    "ComputableData"}];

xmax/ℬ

(* 23.999961539993990714 *)

24 ℬ - xmax

(* 2.5134162337154*10^-6 *)
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  • $\begingroup$ Which is to say, xmax is not really an integer multiple of Norton's constant -- otherwise they would agree to a lot more digits. $\endgroup$
    – ilian
    Jan 6 '18 at 17:58
  • $\begingroup$ Thanks for showing me how to import a constant into Mathematica from Alpha. I figured it wasn't the correct upper bound, I just wanted to learn something new and figured there was a chance. Thanks again. $\endgroup$ Jan 8 '18 at 3:34
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Entering the number above into Wolfram Alpha, the bottom right text makes it very clear what the constant is:

enter image description here

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    $\begingroup$ These are just guesses about a possible closed form, and none of them happens to be correct. $\endgroup$
    – ilian
    Jan 6 '18 at 17:59

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