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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 | ||