Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of $$\mathbb{E}[\exp(X^Tv)]$$ for any $v$?
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1$\begingroup$ Because of the symmetry of the distribution of $X$ the problem is equivalent to determine the function $t \to \mathbb{E}e^{t \cdot X_1}$, $t \in \mathbb{R}$. Of course this does not solve the original problem. $\endgroup$– Dieter KadelkaCommented May 22, 2019 at 23:12
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3$\begingroup$ Since $X^T v = |v| (2 B - 1)$, where $B$ has beta distribution with parameters $(\tfrac{n}{2}, \tfrac{n}{2})$, your problem is equivalent to finding the moment generating function of the beta distribution. If I remember correctly, this is some kind of hypergeometric function; certainly this is well-studied. $\endgroup$– Mateusz KwaśnickiCommented May 22, 2019 at 23:13
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1$\begingroup$ Together with statlect.com/probability-distributions/beta-distribution this problem seems to be solved. $\endgroup$– Dieter KadelkaCommented May 22, 2019 at 23:21
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$\begingroup$ Thank you both! BTW, the beta distribution should be with parameter $(\frac{n-1}{2},\frac{n-1}{2})$ right? $\endgroup$– nevereverneverCommented May 23, 2019 at 15:13
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$\begingroup$ Yes. Especially for $n = 3$ you have $Be(1,1)$, the uniform distribution on $(0,1)$. $\endgroup$– Dieter KadelkaCommented May 23, 2019 at 22:44
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1 Answer
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As Dieter said, this amounts to the function $t\mapsto \mathbb{E}\ e^{tX_1}$. However, instead of $t\in\mathbb{R}$, it is better to look at it with $t\in\mathbb{C}$ since this is an entire analytic function. So modulo replacing $t$ by $it$, this is the same as the Fourier transform of a sphere which is very well known and expressible in terms of Bessel functions.
See the discussion on this MO question: Fourier transform of the unit sphere
answered May 23, 2019 at 17:49