概率论与数理统计 (Spring 2023)/Problem Set 3: Difference between revisions
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== Problem 1 == | == Problem 1 == | ||
<li>[<strong>Moments</strong>] | <li> [<strong>Moments</strong>] | ||
Find an example of a random variable with finite <math>j</math>-th moments for <math>1 \leq j \leq k</math> but an unbounded <math>(k + 1)</math>-th moment. Give a clear argument showing that your choice has these properties. | Find an example of a random variable with finite <math>j</math>-th moments for <math>1 \leq j \leq k</math> but an unbounded <math>(k + 1)</math>-th moment. Give a clear argument showing that your choice has these properties. | ||
</li> | </li> | ||
<li> | <li>[<strong>Moments</strong>] | ||
Let <math>X\sim \text{Geo}(p)</math> be a geometric random variable. Find <math>\mathbb{E}[X^3]</math> and <math>\mathbb{E}[X^4]</math>. (Hint: find a recursive expression from <math>\mathbb{E}[X^{n-1}]</math> to <math>\mathbb{E}[X^{n}]</math> may be useful.) | Let <math>X\sim \text{Geo}(p)</math> be a geometric random variable. Find <math>\mathbb{E}[X^3]</math> and <math>\mathbb{E}[X^4]</math>. (Hint: find a recursive expression from <math>\mathbb{E}[X^{n-1}]</math> to <math>\mathbb{E}[X^{n}]</math> may be useful.) | ||
</li> | </li> | ||
<li>[<strong>Variance</strong>] | <li>[<strong>Variance</strong>] | ||
For pairwise independent random variables <math>X_1,X_2,\cdots, X_n</math>, show that <math>\textbf{Var}\left[\sum_{i=1}^n X_i\right] =\sum_{i=1}^n \textbf{Var} (X_i)</math>. | For pairwise independent random variables <math>X_1,X_2,\cdots, X_n</math>, show that <math>\textbf{Var}\left[\sum_{i=1}^n X_i\right] =\sum_{i=1}^n \textbf{Var} (X_i)</math>. | ||
</li> | </li> | ||
<li> | <li>[<strong>Variance</strong>] | ||
Let <math>X = \sum_{i=1}^N X_i</math>, where <math>X_i (i\geq 1)</math> are independent, identically distributed random variables with mean <math>\mu</math> and variance <math>\sigma^2</math>, and <math>N</math> is positive, integer-valued random variable, and is independent of the <math>X_i</math> for all <math>i\geq 1</math>. Show that <math>\textbf{Var}(X) = \sigma^2\mathbb{E}[N] + \mu^2 \textbf{Var}(N)</math>. | Let <math>X = \sum_{i=1}^N X_i</math>, where <math>X_i (i\geq 1)</math> are independent, identically distributed random variables with mean <math>\mu</math> and variance <math>\sigma^2</math>, and <math>N</math> is positive, integer-valued random variable, and is independent of the <math>X_i</math> for all <math>i\geq 1</math>. Show that <math>\textbf{Var}(X) = \sigma^2\mathbb{E}[N] + \mu^2 \textbf{Var}(N)</math>. | ||
</li> | </li> | ||
<li> | <li>[<strong>Variance</strong>] | ||
Each member of a group of <math>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.) | Each member of a group of <math>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.) | ||
</li> | </li> | ||
<li> | <li>[<strong>Variance</strong>] | ||
An urn contains <math>n</math> balls numbered 1, 2, ..., <math>n</math>. We remove <math>k</math> balls at random (without replacement) and add up their numbers. Find the mean and variance of the sum. | An urn contains <math>n</math> balls numbered 1, 2, ..., <math>n</math>. We remove <math>k</math> balls at random (without replacement) and add up their numbers. Find the mean and variance of the sum. | ||
</li> | </li> | ||
<li>[<strong>Covariance and Correlation</strong>] | |||
<li>[<strong>Covariance and Correlation</strong>] | |||
Let <math>X</math> and <math>Y</math> be discrete random variables with correlation <math>\rho</math>. Show that <math>|\rho|\leq 1</math>. | Let <math>X</math> and <math>Y</math> be discrete random variables with correlation <math>\rho</math>. Show that <math>|\rho|\leq 1</math>. | ||
</li> | </li> | ||
Revision as of 10:05, 24 April 2023
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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.