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cs-401r:assignment-1.5 [2014/09/17 08:34] ringger [Question 2: Independence] |
cs-401r:assignment-1.5 [2014/09/25 19:49] ringger [Question 1: Useful theorems in probability theory] |
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=== Question 1: Useful theorems in probability theory === | === Question 1: Useful theorems in probability theory === | ||

- | [40 points; 20 per sub-problem] | + | [70 points; 35 per sub-problem] |

+ | <html><!-- originally 40, 20 per sub problem --></html> | ||

(Adapted from: Manning & Schuetze, p. 59, exercise 2.1) | (Adapted from: Manning & Schuetze, p. 59, exercise 2.1) | ||

- | Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following statements. Develop your proof first in terms of sets and then translate into probabilities; use set theoretic operations on sets and arithmetic operators on probabilities. Be sure to apply [[Proofs|good proof technique]]: justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification, you must first satisfy the conditions / pre-requisites of the axiom. | + | Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following statements. Develop your proof first in terms of sets and then translate into probabilities; use set theoretic operations on sets and arithmetic operators on probabilities. Be sure to apply [[Proofs|good proof technique]]: justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification, you must first satisfy the conditions / pre-requisites of the axiom. (See the [[example_proofs|example proofs page]] for one example.) |

# $P(\varnothing) = 0$ | # $P(\varnothing) = 0$ | ||

# $A \subseteq B \Rightarrow P(A) \leq P(B)$ | # $A \subseteq B \Rightarrow P(A) \leq P(B)$ | ||

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=== Question 2: Independence === | === Question 2: Independence === | ||

- | [10 points] | + | [15 points] |

+ | <html><!-- originally 10 --></html> | ||

(Adapted from: Manning & Schuetze, p. 59, exercise 2.4) | (Adapted from: Manning & Schuetze, p. 59, exercise 2.4) | ||

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2015: improved wording: If the distribution over two random variables $X$ and $Y$ is defined in the following table, are $X$ and $Y$ independently distributed? | 2015: improved wording: If the distribution over two random variables $X$ and $Y$ is defined in the following table, are $X$ and $Y$ independently distributed? | ||

=== Question 3: Chain rule === | === Question 3: Chain rule === | ||

- | [10 points] | + | [15 points] |

+ | <html><!-- originally 10 --></html> | ||

Prove the chain rule, namely that $P(\cap^n_{i=1} A_i) = P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_1\cap A_2)\cdot ...\cdot P(A_n | \cap^{n-1}_{i=1} A_i)$. Justify each step. Use the same standard of proof as in problem #1 above. | Prove the chain rule, namely that $P(\cap^n_{i=1} A_i) = P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_1\cap A_2)\cdot ...\cdot P(A_n | \cap^{n-1}_{i=1} A_i)$. Justify each step. Use the same standard of proof as in problem #1 above. |