概率论与数理统计 (Spring 2023)/Problem Set 4

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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)

• [Density function] Determine the value of [math]\displaystyle{ C }[/math] such that [math]\displaystyle{ f(x) = C\exp(-x-e^{-x}), x\in \mathbb{R} }[/math] is a probability density function (PDF) for a continuous random variable.
• [Independence] Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be independent and identically distributed continuous random variables with cumulative distribution function (CDF) [math]\displaystyle{ F }[/math] and probability density function (PDF) [math]\displaystyle{ f }[/math]. Find out the density functions of [math]\displaystyle{ V = \max\{X,Y\} }[/math] and [math]\displaystyle{ U = \min\{X,Y\} }[/math].
• [Expectation of random variables] Let [math]\displaystyle{ X }[/math] be a random variable with mean [math]\displaystyle{ \mu }[/math] and continuous cumulative distribution function (CDF) [math]\displaystyle{ F }[/math].
  • (a) Suppose [math]\displaystyle{ X \ge 0 }[/math]. Show that [math]\displaystyle{ \int_{0}^a F(x) dx = \int_{a}^{\infty} [1-F(x)] dx }[/math] if and only if [math]\displaystyle{ a = \mu }[/math].
  • (b) Suppose [math]\displaystyle{ X }[/math] further has finite variance. Show that [math]\displaystyle{ g(a) = \mathbb{E}((X-a)^2) }[/math] is a minimum when [math]\displaystyle{ a = \mu }[/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])