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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.
cs-401r/assignment-1.5.txt · Last modified: 2014/09/25 19:49 by ringger
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