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

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];