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cs-401r:assignment-1 [2014/09/08 08:11] ringger [Question 4: Factoring Joint Distributions] |
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. | ||
| # $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule) | # $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule) | ||
| - | #* Hint: refer to part result #1 as a part of your proof of part #2 | + | #* 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: refer to part result #1 as a part of your proof of part #4 | + | #* 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 $P(X_1, X_2, X_3, X_4, X_5, X_6)$ | + | # How many possible ways can you completely factor the joint probability of six events $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ |
| - | # Given the same joint probability distribution $P(X_1, X_2, X_3, X_4, X_5, X_6)$, 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 5: Conditional Probability === | === Question 5: Conditional Probability === | ||
