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cs-401r:assignment-1.5 [2014/09/24 15:41] cs401rPML [Question 1: Useful theorems in probability theory] added link to example proofs page |
cs-401r:assignment-1.5 [2014/09/25 19:49] (current) ringger [Question 1: Useful theorems in probability theory] |
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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 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]]. | + | Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following 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 [[example_proofs|example proofs page]] for one example.) |

# $P(\varnothing) = 0$ | # $P(\varnothing) = 0$ | ||

# $A \subseteq B \Rightarrow P(A) \leq P(B)$ | # $A \subseteq B \Rightarrow P(A) \leq P(B)$ |