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 cs-401r:assignment-1 [2014/09/05 10:52]cs401rpml cs-401r:assignment-1 [2014/09/24 15:40] (current)cs401rPML [Question 3: Useful theorems in probability theory] added link to example proofs Both sides previous revision Previous revision 2014/09/24 15:40 cs401rPML [Question 3: Useful theorems in probability theory] added link to example proofs2014/09/09 20:56 ringger [Question 3: Useful theorems in probability theory] 2014/09/09 20:55 ringger [Question 4: Factoring Joint Distributions] 2014/09/09 20:55 ringger 2014/09/09 14:53 ringger [Question 4: Factoring Joint Distributions] 2014/09/08 09:23 cs401rpml [Question 1: Warming up] 2014/09/08 09:00 cs401rpml [Question 1: Warming up] 2014/09/08 08:19 ringger [Question 3: Useful theorems in probability theory] 2014/09/08 08:19 ringger [Question 3: Useful theorems in probability theory] 2014/09/08 08:19 ringger [Question 3: Useful theorems in probability theory] 2014/09/08 08:12 ringger [Question 4: Factoring Joint Distributions] 2014/09/08 08:11 ringger [Question 4: Factoring Joint Distributions] 2014/09/05 12:27 ringger [Question 3: Useful theorems in probability theory] 2014/09/05 12:27 ringger [Question 1: Warming up] 2014/09/05 12:26 ringger 2014/09/05 10:52 cs401rpml 2014/09/05 10:46 cs401rpml [Instructions] 2014/09/05 10:46 cs401rpml [Question 3: Factoring Joint Distributions] 2014/09/05 10:45 cs401rpml [Question 2: Joint Probability] 2014/09/05 10:45 cs401rpml [Question 1: Useful theorems in probability theory] 2014/09/05 10:12 ringger 2014/09/05 08:20 ringger 2014/09/05 08:11 ringger 2014/09/03 00:02 ringger created Next revision Previous revision 2014/09/24 15:40 cs401rPML [Question 3: Useful theorems in probability theory] added link to example proofs2014/09/09 20:56 ringger [Question 3: Useful theorems in probability theory] 2014/09/09 20:55 ringger [Question 4: Factoring Joint Distributions] 2014/09/09 20:55 ringger 2014/09/09 14:53 ringger [Question 4: Factoring Joint Distributions] 2014/09/08 09:23 cs401rpml [Question 1: Warming up] 2014/09/08 09:00 cs401rpml [Question 1: Warming up] 2014/09/08 08:19 ringger [Question 3: Useful theorems in probability theory] 2014/09/08 08:19 ringger [Question 3: Useful theorems in probability theory] 2014/09/08 08:19 ringger [Question 3: Useful theorems in probability theory] 2014/09/08 08:12 ringger [Question 4: Factoring Joint Distributions] 2014/09/08 08:11 ringger [Question 4: Factoring Joint Distributions] 2014/09/05 12:27 ringger [Question 3: Useful theorems in probability theory] 2014/09/05 12:27 ringger [Question 1: Warming up] 2014/09/05 12:26 ringger 2014/09/05 10:52 cs401rpml 2014/09/05 10:46 cs401rpml [Instructions] 2014/09/05 10:46 cs401rpml [Question 3: Factoring Joint Distributions] 2014/09/05 10:45 cs401rpml [Question 2: Joint Probability] 2014/09/05 10:45 cs401rpml [Question 1: Useful theorems in probability theory] 2014/09/05 10:12 ringger 2014/09/05 08:20 ringger 2014/09/05 08:11 ringger 2014/09/03 00:02 ringger created Line 16: Line 16: == 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 from: Manning & Schuetze, p. 59, exercise 2.1) + 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 $P(D \cap E)$ ? - 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.   ​ + # What is $P(D|E)$ ? - # $P(B - A) = P(B) - P(A \cap B)$ + # 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 === Line 39: Line 35: ** $P($<​tt>​three-letter-word​$) = 0.0003$ ** $P($<​tt>​three-letter-word​$) = 0.0003$ - === Question 3: Factoring Joint Distributions ​=== + === 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 4: Factoring Joint Probabilities ​=== [10 points] [10 points] - # How many possible ways can you completely factor the joint distribution ​$P(X_1, X_2, X_3, X_4, X_5, X_6)$ + # How many possible ways can you completely factor the joint probability of six events ​$P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ - # For some arbitrary ​joint probability ​distribution on six random variables ​$P(X_1, X_2, X_3, X_4, X_5, X_6)$, 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 ​4: Conditional Probability === + === Question ​5: Conditional Probability === [10 points] [10 points]