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cs-401r:assignment-1 [2014/09/09 20:55]
ringger [Question 4: Factoring Joint Distributions]
cs-401r:assignment-1 [2014/09/24 15:40]
cs401rPML [Question 3: Useful theorems in probability theory] added link to example proofs
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 (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 five 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 five 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 proofs on the [[example_proofs|example proofs page]].
 # $P(B - A) = P(B) - P(A \cap B)$ # $P(B - A) = P(B) - P(A \cap B)$
 #* Note that inside the $P(\cdot)$, the '​$-$'​ operator indicates set difference. #* Note that inside the $P(\cdot)$, the '​$-$'​ operator indicates set difference.
 # $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule) # $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule)
 #* Hint: use the theorem in part #1 as a step in your proof #* Hint: use the theorem in part #1 as a step in your proof
-# $P(\neg A) = 1 - P(A)$+# $P(\overline{A}) = 1 - P(A)$
 #* Hint: use the theorem in part #1 as a step in your proof #* Hint: use the theorem in part #1 as a step in your proof
  
cs-401r/assignment-1.txt ยท Last modified: 2014/09/24 15:40 by cs401rPML
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