#heavy-tails

Question 1: Single choice

To check that the standard Cauchy density $p(x) = \frac{1}{\pi(1 + x^2)}$ integrates to one, a friend uses importance sampling with a standard normal proposal $q$ : $\int p(x)\,dx \approx \frac{1}{n}\sum_{i=1}^n p(x_i)/q(x_i)$ , $x_i \sim q$ . Two independent runs give clearly different values, and within each run the estimate jumps around as $n$ grows. Why?

The estimator is biased, because the proposal does not cover the same support as $p$ 1
The sample sizes are just too small; any run with enough samples settles down quickly2
Importance sampling cannot estimate normalising constants at all3
The weights $p/q$ have infinite variance: normal tails are far lighter than Cauchy tails4

Timed test