What's the expectation of the function given by $|X_1|$ when $X=(X_1 \dots X_d) \sim Unif(S^{d-1}(r))$

Question

Let $X=(X_1 \dots X_d) \sim Unif(r S^{d-1} \equiv S^{d-1}(r) ).$ I'd like to know what's the expectation of $|X_1|,$ i.e. what's the integral:

$$ \frac{1}{vol(S^{d-1}(r))} \int_{S^{d-1}(r)} |x_1| d vol_{S^{d-1}(r)}? $$

I think my problem is with the complicated volume form in high dimensional spheres, and reducing the polar coordinates to cartesian co-ordinates for integration. This is where I need a bit of help, unless there's more slick way using symmetry etc.

Thanks in advance!!

Answer 1

Too long for a comment.

I am far from an expert, but would this approach work? If you sum up all the $d Vol(S^{d-1}(r))$ pieces where $X_1 \in [x_1, x_1 + d x_1]$, the total $(d-1)$-dimensional volume would be:

$$d x_1 \times Vol(S^{d-2}(\sqrt{r^2 - x_1^2}))$$

So your integral becomes

$$\int_{-r}^r |x_1| \, Vol(S^{d-2}(\sqrt{r^2 - x_1^2}))\, d x_1$$

And you can plug in the known formula for $Vol(S^{d-2})$. It may still be difficult to actually integrate, but at least it is now just an integration of one real variable $x_1$.