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cs-401r:assignment-1 [2014/09/05 10:45]
cs401rpml [Question 1: 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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-'''​Probability Theory'''​+Probability Theory ​=
  
 == Objectives == == Objectives ==
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 == Instructions == == Instructions ==
  
-This is a mathematical homework assignment. Show your work. Be clear and concise. ​ Type your assignment.<br>+This is a mathematical homework assignment. Show your work. Be clear and concise. ​ Type your assignment.
  
 I strongly recommend you work through these exercises as soon as possible for their instructional content. If you have general questions about the assignment, please post on the Google Group. ​ Finish early, and earn the early bonus. I strongly recommend you work through these exercises as soon as possible for their instructional content. If you have general questions about the assignment, please post on the Google Group. ​ Finish early, and earn the early bonus.
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 == Exercises == == Exercises ==
  
-=== Question 1: Useful theorems in probability theory ​=== +=== Question 1: Warming up === 
-[30 points; ​10 per sub‐problem]+[10 points: 2 for the first, 3 each for the second and third parts]
  
-(Adapted fromManning & Schuetze, p. 59, exercise 2.1) +Let's begin with the same sample space $\Omega$ we have been using in classLet $\Omega$ 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 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 $P(\cap E)$ ? 
-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 justification on the rightRemember that in order to invoke an axiom as justification,​ you must first satisfy ​the conditions / pre-requisites ​of the axiom  ​ +What is $P(D|E)$ ? 
-# $P(B - AP(BP(\cap B)$ +What is $P(E|D)$ ?
-# $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule) +
-#* Hint: refer to part result #1 as a part of your proof of part #2 +
-# $P(\neg A) = 1 - P(A)$ +
-#* Hint: refer to part result #1 as a part of your proof of part #4+
  
 === Question 2: Joint Probability === === Question 2: Joint Probability ===
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 * Let <​tt>​is-abbreviation</​tt>​ denote the event "this period indicates an abbreviation",​ and let <​tt>​three-letter-word</​tt>​ denote "a period occurs after a three letter word" * Let <​tt>​is-abbreviation</​tt>​ denote the event "this period indicates an abbreviation",​ and let <​tt>​three-letter-word</​tt>​ denote "a period occurs after a three letter word"
 * Assume the following probabilities:​ * Assume the following probabilities:​
-** <​math>​P(</​math>​<​tt>​is-abbreviation</​tt>​<​math> ​</​math>​<​tt>​three-letter-word</​tt>​<​math>​) = 0.8</​math>​ +** $P($<​tt>​is-abbreviation</​tt>​$<​tt>​three-letter-word</​tt>​$) = 0.8$ 
-** <​math>​P(</​math>​<​tt>​three-letter-word</​tt>​<​math>​) = 0.0003</math>+** $P($<​tt>​three-letter-word</​tt>​$) = 0.0003
 + 
 +=== Question 3: Useful theorems in probability theory === 
 +[60 points; 20 per sub‐problem] 
 + 
 +(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. See the proofs on the [[example_proofs|example proofs page]]. 
 +# $P(B - A) = P(B) - P(A \cap B)$ 
 +#* 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) 
 +#* Hint: use the theorem in part #1 as a step in your proof 
 +# $P(\overline{A}) = 1 - P(A)$ 
 +#* Hint: use the theorem in part #1 as a step in your proof
  
-=== Question ​3: Factoring Joint Distributions ​===+=== Question ​4: Factoring Joint Probabilities ​===
 [10 points] [10 points]
  
-# How many possible ways can you completely factor the joint distribution <​math>​P(X_1,​ X_2, X_3, X_4, X_5, X_6)</​math>?​ +# 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)
-# For some arbitrary ​joint probability ​distribution on six random variables <​math>​P(X_1, X_2, X_3, X_4, X_5, X_6)</​math>,​ 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 ​4: Conditional Probability ===+=== Question ​5: Conditional Probability ===
 [10 points] [10 points]
  
cs-401r/assignment-1.1409935510.txt.gz · Last modified: 2014/09/05 10:45 by cs401rpml
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