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

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== Problem 1 ==
== Problem 1 ==
 
<ul>
<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.
{{Theorem|''Algorithm''|
{{Theorem|''Algorithm''|
:'''Input:'''  real numbers <math>U < 1</math>;
:'''Input:'''  real numbers <math>U < 1</math>;
----
----
:
:initialize <math>x = 1</math> and <math>count = 0</math>;
:initialize <math>x = 1</math> and <math>count = 0</math>;
:while <math> x > U </math> do
:while <math> x > U </math> do
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:return <math>count</math>;
:return <math>count</math>;
}}
}}
</li>
</ul>

Revision as of 08:49, 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

  • [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];