概率论与数理统计 (Spring 2023)/Problem Set 3

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Assumption throughout Problem Set 3

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 (Warm-up Problems)

• [Variance (I)] Let [math]\displaystyle{ X_1,X_2,\cdots, X_n }[/math] be pairwise independent random variables. Show that [math]\displaystyle{ \textbf{Var}\left[\sum_{i=1}^n X_i\right] =\sum_{i=1}^n \textbf{Var} [X_i] }[/math].
• [Variance (II)] Let [math]\displaystyle{ X = \sum_{i=1}^N X_i }[/math], where [math]\displaystyle{ (X_i)_{i \ge 1} }[/math] are independent, identically distributed random variables with mean [math]\displaystyle{ \mu }[/math] and variance [math]\displaystyle{ \sigma^2 }[/math], and [math]\displaystyle{ N }[/math] is positive, integer-valued random variable, and is independent of the [math]\displaystyle{ X_i }[/math] for all [math]\displaystyle{ i\geq 1 }[/math]. Show that [math]\displaystyle{ \textbf{Var}[X] = \sigma^2\mathbb{E}[N] + \mu^2 \textbf{Var}[N] }[/math].
• [Variance (III)] Each member of a group of [math]\displaystyle{ n }[/math] players rolls a dice. For any pair of players who throw the same number, the group scores [math]\displaystyle{ 1 }[/math] point. Find the mean and variance of the total score of the group.
• [Variance (IV)] An urn contains [math]\displaystyle{ n }[/math] balls numbered [math]\displaystyle{ 1, 2, \ldots, n }[/math]. We select [math]\displaystyle{ k }[/math] balls uniformly at random without replacement and add up their numbers. Find the mean and variance of the sum.
• [Moments (I)] Find an example of a random variable with finite [math]\displaystyle{ j }[/math]-th moments for [math]\displaystyle{ 1 \leq j \leq k }[/math] but an unbounded [math]\displaystyle{ (k + 1) }[/math]-th moment. Give a clear argument showing that your choice has these properties.
• [Moments (II)] Let [math]\displaystyle{ X\sim \text{Geo}(p) }[/math] for some [math]\displaystyle{ p \in (0,1) }[/math]. Find [math]\displaystyle{ \mathbb{E}[X^3] }[/math] and [math]\displaystyle{ \mathbb{E}[X^4] }[/math].
• [Moments (III)] Let [math]\displaystyle{ X\sim \text{Pois}(\lambda) }[/math] for some [math]\displaystyle{ \lambda \gt 0 }[/math]. Find [math]\displaystyle{ \mathbb{E}[X^3] }[/math] and [math]\displaystyle{ \mathbb{E}[X^4] }[/math].
• [Covariance and correlation (I)] Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be discrete random variables with correlation [math]\displaystyle{ \rho }[/math]. Show that [math]\displaystyle{ |\rho|\leq 1 }[/math].
• [Covariance and correlation (II)] Let [math]X[/math] and [math]Y[/math] be discrete random variables with mean 0, variance 1, and correlation [math]\rho[/math]. Show that [math]\mathbb{E}(\max\{X^2,Y^2\})\leq 1+\sqrt{1-\rho^2}[/math]. (Hint: use the identity [math]\max\{a,b\} = \frac{1}{2}(a+b+|a-b|)[/math].)
• [Covariance and correlation (III)] Construct two random variables [math]X[/math] and [math]Y[/math] such that their covariance [math]\textbf{Cov}(X,Y) = 0[/math] but [math]X[/math] and [math]Y[/math] are not independent. You should proof your construction is true.