Rejection-sampling question, stuck in math

Solution 1:

Each time we pick a point in $[-1,1]^d$ is one Bernoulli trial where the success criterion is that the point is in $B.$ The chance of success is $p = \mathbb P(x\in B).$ You will do some number $Y$ trials up to and including the first trial that is a success, then you will output the point found on the $Y$th trial. The number $Y$ is a random variable with an exponential distribution and mean $1/p.$

The exponential distribution is well-known, as are its mean and its relationship to repeated Bernoulli trials. You should be able to find these things explained in most decent probability textbooks, for example here.

Quickly summarizing: on each attempt you have probability $p$ to pick a point $x\in B,$ probability $1-p$ to pick a point you can't use. The probability that it takes exactly $n$ trials to pick a suitable $x$ is the probability that you fail exactly $n-1$ times in a row and then succeed,

$$ \mathbb P(Y=n) = (1-p)^{n-1} p. $$

We get the expected value in the usual way for a variable $Y$ that can take on values $1, 2, 3, \ldots$:

$$ \mathbb E(Y) = \sum_{n=1}^\infty n \mathbb P(Y=n). $$

After working out the series, we find $\mathbb E(Y) = 1/p.$

A more intuitive way to visualize this result is that we have on average $p$ successes per attempt, or one success for every $1/p$ attempts, that is, on average $1/p$ attempts per success.

Somehow I thought I'd be able to just link to another answer with all of this but did not have good luck.