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

[Moments]

1. 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.
2. Let [math]\displaystyle{ X\sim \text{Geo}(p) }[/math] be a geometric random variable. Find [math]\displaystyle{ \mathbb{E}[X^3] }[/math] and [math]\displaystyle{ \mathbb{E}[X^4] }[/math]. (Hint: find a recursive expression from [math]\displaystyle{ \mathbb{E}[X^{n-1}] }[/math] to [math]\displaystyle{ \mathbb{E}[X^{n}] }[/math] may be useful.)

[Variance]

1. For pairwise independent random variables [math]\displaystyle{ X_1,X_2,\cdots, X_n }[/math], show that [math]\displaystyle{ \textbf{Var}\left[\sum_{i=1}^n X_i\right] =\sum_{i=1}^n \textbf{Var} (X_i) }[/math].
2. Let [math]\displaystyle{ X = \sum_{i=1}^N X_i }[/math], where [math]\displaystyle{ X_i (i\geq 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].
3. Each member of a group of [math]\displaystyle{ n }[/math] players rolls a dice (with six faces). For any pair of players who throw the same number, the group scores 1 point. Find the mean and variance of the total score of the group. (Hint: use the property of pairwise independent.)
4. An urn contains [math]\displaystyle{ n }[/math] balls numbered 1, 2, ..., [math]\displaystyle{ n }[/math]. We remove [math]\displaystyle{ k }[/math] balls at random (without replacement) and add up their numbers. Find the mean and variance of the sum.

[Covariance and Correlation]

1. 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].