# From built-in symbols to algebraic representation

If I have an expression with Mathematica's built-in functions like

Probability[x > 0, x \[Distributed] UniformDistribution[{-1, 1}]]


its algebraic representation would be something like

Integrate[PDF[UniformDistribution[{-1, 1}], x], {x, 0, \[Infinity]}]


If I understand correctly, Mathematica's engine does the transformation from first to second representation and then evaluates it.

Is there a function that will take as input first expression and give me the second?

• It does not appear that Mathematica takes the straightforward path. Evaluate Probability[x > 0, x \[Distributed] UniformDistribution[{-1, 1}]] // Trace Jan 15, 2016 at 14:04
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• @BobHanlon Ok thanks so it's much more complex than I imagined. Maybe what I'm asking is not possible, or just in some very simple cases.
– Gleb
Jan 15, 2016 at 14:12
• Sometimes these things are done by table lookup. That seems likely with common, simple distributions like UniformDistribution. Jan 15, 2016 at 14:18

You could use pattern-matching on the HoldAll form of the Probability input and transfer from the first form to the second. The three arguments are the predicate, variable, and the distribution. So for your example

{pred, var, dist} = {x>0, x, UniformDistribution[{-1, 1}]}


the equivalence is

Probability[pred, Distributed[var, dist] ==
Integrate[Boole[pred] PDF[dist, x], {x, -Infinity, Infinity}]

• Thanks a lot! That seems to answer my question :)
– Gleb
Jan 20, 2016 at 12:17