数据科学基础 (Fall 2024)/Problem Set 5

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Assumption throughout Problem Set 5

Without further notice, we are working on probability space [math]\displaystyle{ (\Omega,\mathcal{F},\Pr) }[/math].

Without further notice, we assume that the expectation of random variables are well-defined.

Problems (Continuous Random Variables)

• [Distribution function] Can an [math]\displaystyle{ F:\mathbb R\to [0,1] }[/math], which is (i) nondecreasing, (ii) [math]\displaystyle{ \lim_{x\to-\infty}F(x)=0,\lim_{x\to+\infty}F(x)=1 }[/math], (iii) continuous, (iv) not differentiable at some point, be a cdf for some random variable? Is $F$ always a cumulative distribution function for some random variable? What random variable might it be? What if [math]\displaystyle{ F }[/math] is not differentiable at every point? Justify your answer.
• [Density function] For what values of the [math]\displaystyle{ C }[/math], [math]\displaystyle{ f (x)= C{x(1− x)}^{−1/2} , 0 \lt x \lt 1, }[/math], the density function of the ‘arc sine law’, is a probability density function?
• [Max and min] Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be independent random variables with common distribution function [math]\displaystyle{ F }[/math] and density function [math]\displaystyle{ f }[/math]. Show that [math]\displaystyle{ V= \max\{X,Y\} }[/math] has distribution function [math]\displaystyle{ \Pr(V \le x)= F(x)^2 }[/math] and density function [math]\displaystyle{ f_V (x)= 2 f (x)F(x), x \in \mathbb R }[/math]. Find the density function of [math]\displaystyle{ U= \min\{X,Y \} }[/math].
• [Expectation] Let [math]\displaystyle{ X, Y }[/math] be two independent and identically distributed continuous random variables with cumulative distribution function (CDF) [math]\displaystyle{ F }[/math]. Furthermore, [math]\displaystyle{ X,Y \ge 0 }[/math]. Show that [math]\displaystyle{ \mathbb{E}[|X-Y|] = 2 \left(\mathbb{E}[X] - \int_{0}^{\infty} (1-F(x))^2 dx\right) }[/math]
• [Exponential distribution (i)] Prove that exponential distribution is the only memoryless continuous random variable.
• [Exponential distribution (ii)] Let [math]\displaystyle{ U }[/math] and [math]\displaystyle{ X }[/math] be independent, where [math]\displaystyle{ U }[/math] is uniform on [math]\displaystyle{ (0,1) }[/math] and [math]\displaystyle{ X }[/math] is exponentially distributed with parameter [math]\displaystyle{ \lambda }[/math]. Show that [math]\displaystyle{ \mathbb E(\min\{U,X\})= \lambda^{−1}e^{−\lambda} − \lambda^{−2}(1− e^{−\lambda}) }[/math].
• [Geometric distribution] Prove that [math]\displaystyle{ \lfloor X\rfloor }[/math] is a geometric random variable, and find its probability mass function, where [math]\displaystyle{ X\sim\exp(\lambda) }[/math].
• [Poisson clocks] Prove that a Poisson point process with [math]\displaystyle{ k }[/math] Poisson clocks with rate [math]\displaystyle{ \lambda }[/math] is equivalent to the [math]\displaystyle{ 1 }[/math]-clock process with rate [math]\displaystyle{ \lambda k }[/math].
• [Marginal normal distributions] Prove that the marginal distributions of a joint normal distribution is a normal distribution.