Generate Uniform Samples and Estimate Percentiles

Answer the following two probability/statistics questions. 1. You are given access to a function `rand01()` that returns independent samples from `Uniform(0, 1)`. Write a function `sample_square()` that returns a 2D point `(x, y)` uniformly distributed over the square `(-1, 1) x (-1, 1)`. 2. You are given an aggregated histogram for a numeric variable related to search queries. Each bucket `i` contains: - `left_bd` (float): left boundary of the bucket - `right_bd` (float): right boundary of the bucket - `search_count` (int): number of observations in that bucket There are `K` ordered, non-overlapping buckets, and you do not have access to the raw observations. Describe how to estimate the `p`-th percentile, where `p` is between `0` and `100`. First discuss a naive approach based on the bucket that contains the percentile, then propose a better approximation that uses interpolation within that bucket. State any assumptions your method relies on.