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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] Both sides previous revision Previous revision 2014/09/25 19:49 ringger [Question 1: Useful theorems in probability theory] 2014/09/24 15:41 cs401rPML [Question 1: Useful theorems in probability theory] added link to example proofs page2014/09/18 17:41 cs401rPML Updated points to reflect how scored.2014/09/17 08:34 ringger [Question 2: Independence] 2014/09/10 11:49 ringger [Question 2: Independence] 2014/09/10 11:46 ringger [Question 1: Useful theorems in probability theory] 2014/09/06 23:27 ringger [Question 1: Useful theorems in probability theory] 2014/09/05 11:14 cs401rpml [Question 2: Independence] 2014/09/05 11:14 cs401rpml [Question 2: Independence] 2014/09/05 11:05 cs401rpml [Question 2: Independence] 2014/09/05 11:05 cs401rpml [Question 2: Independence] 2014/09/05 11:04 cs401rpml [Question 3: Chain rule] 2014/09/05 11:04 cs401rpml [Question 2: Independence] 2014/09/05 11:03 cs401rpml [Question 1: Useful theorems in probability theory] 2014/09/05 11:02 cs401rpml [Instructions] 2014/09/05 11:02 cs401rpml 2014/09/05 08:22 ringger created Next revision Previous revision 2014/09/25 19:49 ringger [Question 1: Useful theorems in probability theory] 2014/09/24 15:41 cs401rPML [Question 1: Useful theorems in probability theory] added link to example proofs page2014/09/18 17:41 cs401rPML Updated points to reflect how scored.2014/09/17 08:34 ringger [Question 2: Independence] 2014/09/10 11:49 ringger [Question 2: Independence] 2014/09/10 11:46 ringger [Question 1: Useful theorems in probability theory] 2014/09/06 23:27 ringger [Question 1: Useful theorems in probability theory] 2014/09/05 11:14 cs401rpml [Question 2: Independence] 2014/09/05 11:14 cs401rpml [Question 2: Independence] 2014/09/05 11:05 cs401rpml [Question 2: Independence] 2014/09/05 11:05 cs401rpml [Question 2: Independence] 2014/09/05 11:04 cs401rpml [Question 3: Chain rule] 2014/09/05 11:04 cs401rpml [Question 2: Independence] 2014/09/05 11:03 cs401rpml [Question 1: Useful theorems in probability theory] 2014/09/05 11:02 cs401rpml [Instructions] 2014/09/05 11:02 cs401rpml 2014/09/05 08:22 ringger created Line 17: Line 17: === 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 -->​ (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.) # $P(\varnothing) = 0$ # $P(\varnothing) = 0$ # $A \subseteq B \Rightarrow P(A) \leq P(B)$ # $A \subseteq B \Rightarrow P(A) \leq P(B)$ Line 27: Line 28: === Question 2: Independence === === Question 2: Independence === - [10 points] + [15 points] + <​html><​!-- originally 10 -->​ (Adapted from: Manning & Schuetze, p. 59, exercise 2.4) (Adapted from: Manning & Schuetze, p. 59, exercise 2.4) Line 39: Line 41: 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 -->​ 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.