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 | ||
| Line 24: | Line 25: | ||
: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];