appm4570-unit-2-probability-annotated (1).pdf

appm4570-unit-2-probability-annotated (1).pdf - Unit#2...

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Unit #2: Probability Sections 3.1 - 3.3 1
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Learning Objectives At the end of this unit, students should be able to: 1. Define the terms sample space , outcome , and event (simple and compound). 2. Write out the axioms of probability theory. 3. Prove basic theorems of probability theory. 4. Calculate the probability of events arising from simple experiments (e.g., coin flipping, dice rolling, simple real-world examples) and from real-world examples. This will require the ability to perform basic counting tasks (e.g., permutations and combinations). 5. Define a random variable and calculate the probability that a random variable takes on a particular value. 6. Articulate the importance of an interpretation of probability and identify the two main interpretations. 7. Define conditional probability and calculate conditional probabilities. 8. Apply the multiplication rule. 9. Define independence, and calculate the probability of independent events. 10.Prove Bayes’ theorem and the Law of Total Probability; use them in applications. 11.Use R to estimate probabilities using simulations. 2
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What is Probability? 3
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What is Probability? One main objective of statistics/data science is to help make good decisions under conditions of uncertainty. Examples? Probability is one way to quantify outcomes that cannot be predicted with certainty. 4 In studying a disease , we might want to Know how prevalent it Is in the pop . We May learn this by studying a sample . The inference from sample to pop . Is Uncertain . T prob . theory .
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Sample Space Definition: A probabilistic process is system/ experiment whose outcomes are uncertain. Definition: An outcome is a possible result of a probabilistic process . Definition : A sample space of a probabilistic process is the set of all possible outcomes of that process. 5
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Setty . A set Is a collection of objects . Notation : A = { a , , az , ... , an } ( a , b ) , a and b are # s . . 1 Al = size / # of elements Ex : MY = { 0/1,2 , 3 , ... } . IN l=a . R = all real # s . |R|=a . z =/ ... -2 , -1,0 , I , 2 . - - } Q , e Ex . A = { 1) 2 , 3,4 } A =D Xp = { 1 , 2 , 2 , 3.4 }
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Setoperations . Union : A u B = all elements in A or in B [email protected]@ B r . Intersection : A n B = all elements in A and B . .com#t : A ' = all elements not in A ( but In r )
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Sample Space Sample spaces for: Tossing a coin Selecting a card from a deck Measuring your commute time on a particular morning 6 r ={ H ,t } . 1rl=2 r={ 252,24 , . . . , 1092,108 , ... , JSZ , ... , KSZ , ... } , 111=52 13 ranks , 4 suits . r = ( 0 , T ) , TER
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Events Definition : An event is any collection (subset) of outcomes from the sample space . An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome. When an experiment is performed, an event A is said to occur if the resulting experimental outcome is contained in A.
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