概率论与数理统计 (Spring 2023)/Problem Set 4

• 目前作业非最终版本！

• 每道题目的解答都要有完整的解题过程，中英文不限。

• 我们推荐大家使用LaTeX, markdown等对作业进行排版。

Assumption throughout Problem Set 4

Without further notice, we are working on probability space [math]\displaystyle{ (\Omega,\mathcal{F},\mathbf{Pr}) }[/math].

Without further notice, we assume that the expectation of random variables are well-defined.

The term [math]\displaystyle{ \log }[/math] used in this context refers to the natural logarithm.

Problem 1 (Continuous Random Variables, 30 points)

• Let [math]\displaystyle{ f(x) = C\exp(-x-e^{-x}), x\in \mathbb{R} }[/math] be a probability density function, find the value of [math]\displaystyle{ C }[/math].
• [Random Process] Given a real number [math]\displaystyle{ U\lt 1 }[/math] as input of the following process, find out the expected returning value.

  Algorithm

  Input: real numbers [math]\displaystyle{ U \lt 1 }[/math]; initialize [math]\displaystyle{ x = 1 }[/math] and [math]\displaystyle{ count = 0 }[/math]; while [math]\displaystyle{ x \gt U }[/math] do choose [math]\displaystyle{ y \in (0,1) }[/math] uniformly at random; update [math]\displaystyle{ x = x * y }[/math] and [math]\displaystyle{ count = count + 1 }[/math]; return [math]\displaystyle{ count }[/math];

• [Random Semicircle] We sample [math]\displaystyle{ n }[/math] points within a circle [math]\displaystyle{ C=\{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 \le 1\} }[/math] independently and uniformly at random (i.e., the density function [math]\displaystyle{ f(x,y) \propto 1_{(x,y) \in C} }[/math]). Find out the probability that they all lie within some semicircle with radius [math]\displaystyle{ 1 }[/math]. (Hint: you may apply the technique of change of variables, see function of random variables or Chapter 4.7 in [GS])