Interesting riddle about heights

Question

Choose a person randomly on the street, $X$. Let $N$ denote the random variable representing the number of people that you select randomly from the street before you find someone who's taller than $X$. What is $E[N]$?

Answer 1

Surprisingly, the expected number is infinity.

Suppose that hights have a PDF $f_X$ and CDF $F_X$.

Given that the height of the first person is $x_0$, the probability of each other person (assuming iid) to be taller is $1-F_X(x_0)$. The number of people that should be measured before we find someone taller is geometric, so $E(N\vert X=x_0)=\tfrac{1}{1-F_X(x_0)}$.

Using total expectation, we get $$E(N)=E(E(N\vert X))=\int\limits_{-\infty}^{\infty} \tfrac{f_x(x)dx}{1-F_X(x_0)}=-\ln (1-F_X(x))\vert_{-\infty}^{\infty}=\infty$$

A nice way to see it is to assume that $X$ has a uniform distribution over $(0,1)$. The calculation is similar to the above but with simple functions and the result is the same. This assumption is "allowed" as in the question there is no assumption regarding the distribution of $X$, so it is possible to deduce that this should not matter, and in this case - we can test a simple distribution to get intuition.