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cs-401r:assignment-1 [2014/09/08 08:19]
ringger [Question 3: Useful theorems in probability theory]
cs-401r:assignment-1 [2014/09/24 15:40] (current)
cs401rPML [Question 3: Useful theorems in probability theory] added link to example proofs
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 === Question 1: Warming up === === Question 1: Warming up ===
-[10 points: 2 for the first, 3 each for the second and thirs parts]+[10 points: 2 for the first, 3 each for the second and third parts]
  
-Let's begin with the same sample space $\Omega$ we have been using in class: Let $\Omega$ be the set of all outcomes defined by three successive coin flips. ​ Assume that the coin is fair and that the distribution over all singleton events (events containing one outcome) is uniform. ​ Let the event $D$ be the set of all outcomes in which the second flip is a head.  Let the event $E$ be the set of all outcomes in which teh final flip is a tail. +Let's begin with the same sample space $\Omega$ we have been using in class: Let $\Omega$ be the set of all outcomes defined by three successive coin flips. ​ Assume that the coin is fair and that the distribution over all singleton events (events containing one outcome) is uniform. ​ Let the event $D$ be the set of all outcomes in which the second flip is a head.  Let the event $E$ be the set of all outcomes in which the final flip is a tail. 
 # What are $P(D)$, $P(E)$, and $P(D \cap E)$ ? # What are $P(D)$, $P(E)$, and $P(D \cap E)$ ?
 # What is $P(D|E)$ ? # What is $P(D|E)$ ?
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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
  
-=== Question 4: Factoring Joint Distributions ​===+=== Question 4: Factoring Joint Probabilities ​===
 [10 points] [10 points]
  
-# How many possible ways can you completely factor ​an arbitrary ​joint probability ​distribution on six events $P(A_1A_2A_3A_4A_5A_6)$ +# How many possible ways can you completely factor ​the joint probability ​of six events $P(A_1 ​\cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ 
-Given the same joint probability ​distribution ​$P(A_1A_2A_3A_4A_5A_6)$, apply the chain rule to completely factor this distribution ​in one way.+Apply the chain rule to completely factor ​the joint probability $P(A_1 ​\cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ in one way.
  
 === Question 5: Conditional Probability === === Question 5: Conditional Probability ===
cs-401r/assignment-1.1410185999.txt.gz · Last modified: 2014/09/08 08:19 by ringger
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