For this question, let us define "[closed form](https://brilliant.org/wiki/closed-form-expressions/)" as an expression restricted to addition, subtraction, multiplication, and division; exponents and logarithms, including $e^x$ and $\ln{x}$; trigonometric functions and inverse trigonometric functions. Using an infinite number of terms or operations is not permitted.

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> True or false: For continuous random variable $X$, if $E(X)$ has a closed form, then $P(X<E(X))$ has a closed form.
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I don't see why this would be true, but I can't think of a counter-example.

**Context**: $(1+u\_1)(1+u\_1 u\_2)(1+u\_1 u\_2 u\_3)...$, where the $u$'s are i.i.d. $\text{Uniform}(0,1)$-variables, has a closed form [expectation](https://math.stackexchange.com/q/4644052/398708), $e$. I have tried unsuccessfully to find the [probability that it is greater than its expectation](https://math.stackexchange.com/q/4644230/398708). I wonder, since the expectation has a closed form, should I, or shouldn't I, expect a closed form for the probability? (My definition of closed form does not allow limits, but admits $e$ as a closed form; this is [not a contradiction](https://math.stackexchange.com/q/4644830/398708).)