概率论与数理统计 (Spring 2023)/Problem Set 4: Difference between revisions

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<p>The term <math>\log</math> used in this context refers to the natural logarithm.</p>
<p>The term <math>\log</math> used in this context refers to the natural logarithm.</p>


== Problem 1 ==
== Problem 1 (Continuous Random Variables, 30 points)==
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<ul>
<li>
Let <math>f(x) = C\exp(-x-e^{-x}), x\in \mathbb{R}</math> be a probability density function, find the value of <math>C</math>.
</li>
<li>[<strong>Random Process</strong>]
<li>[<strong>Random Process</strong>]
Given a real number <math>U<1</math> as input of the following process, find out the expected returning value.
Given a real number <math>U<1</math> as input of the following process, find out the expected returning value.
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</li>
</li>
<li>
<li>
[<strong>Random Semicircle</strong>] We sample <math>n</math> points within a circle <math>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>f(x,y) \propto 1_{(x,y) \in C}</math>). Find out the probability that they all lie within some semicircle with radius <math>1</math>. (Hint: you may apply the technique of change of variables, see [https://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables function of random variables] or Chapter 4.7 in )
[<strong>Random Semicircle</strong>] We sample <math>n</math> points within a circle <math>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>f(x,y) \propto 1_{(x,y) \in C}</math>). Find out the probability that they all lie within some semicircle with radius <math>1</math>. (Hint: you may apply the technique of change of variables, see [https://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables function of random variables] or Chapter 4.7 in [GS])
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</ul>
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Revision as of 09:17, 22 May 2023

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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)

  • 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])