371chapter1f2011

371chapter1f2011 - 0 Chapter 1 Probability 1.1 Getting...

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Chapter 1 Probability 1.1 Getting Started These notes will explore how the discipline of Statistics helps scientists learn about the world. There are two main areas in which Statistics helps: Validity. Proper use of Statistics helps a scientist learn things that are true; improper use can lead to a scientist learning things that are false. EfFciency. Proper use of Statistics can help a scientist learn faster ,orwithless effort or at a lower cost . The Frst tool you need to become a good statistician is a cursory understanding of probability. Ibeginwithoneofmyfavoritequotesfromafavoritesource . Predictions are tough, especially about the future.—Yogi Berra. Probability theory is used by mathematicians, scientists and statisticians to quantify uncertainty about the future. We begin with the notion of a chance mechanism .Th isisatwo-wordtechn ica lexpress ion .I t is very important that we use technical expressions exactly as they are deFned. In every day life you may have several meanings for some of your favorite words,forexamplephat,butinthisclass technical expressions have a unique meaning. In these notes the Frst occurrence of a technical expression/term will be in bold-faced type. Both words in ‘chance mechanism’ (CM) are meaningful. The second word reminds us that the CM, when operated ,producesan outcome .TheFrs twordrem indsustha ttheou tcome cannot be predicted with certainty . Several examples will help. 1. CM: A coin is tossed. Outcome: The face that lands up, eitherheadsortails. 2. CM: A (six-sided) die is cast. Outcome: The face that lands up, either 1, 2, 3, 4, 5 or 6. 1
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3. CM: A man with AB blood and a woman with AB blood have a child. Outcome: The blood type of the child, either A, B or AB. 4. CM: The next NFL Super Bowl game. Outcome: The winner of the game, which could be any one of the 32 NFL teams. The next idea is the sample space ,usua l lydeno tedby S .Thesamp lespa c ei stheco l le c t ion of all possible outcomes of the CM. Below are the sample spacesforeachCMlistedabove . 1. CM: Coin. S = { H,T } . 2. CM: Die. S = { 1 , 2 , 3 , 4 , 5 , 6 } . 3. CM: Blood. S = { A, B, AB } . 4. CM: Super Bowl. S = Alistofthe32NFLteams. An event is a collection of outcomes; that is, it is a subset of the sample space. Events are typically denoted by upper case letters, usually from the beginning of the alphabet. Below are some events for each CM listed above. 1. CM: Coin. A = { H } , B = { T } . 2. CM: Die. A = { 5 , 6 } , B = { 1 , 3 , 5 } . 3. CM: Blood. C = { A, B } . 4. CM: Super Bowl. A = { Vikings, Packers, Bears, Lions } . Sometimes it is convenient to describe an event with words. Asexamplesofthis:Forthedie , event A can described as ‘the outcome is larger than 4,’ and event B can be described as ‘the outcome is an odd integer.’ For the Super Bowl, event A can described as ‘the winner is from the NFC North Division.’ Here is where I am going with this: Before aCMisoperated,nobodyknowswhattheoutcome will be. In particular, for any event A that is not the entire sample space, we don’t know whether the outcome will be a member of A . After the CM is operated we can determine/see whether the
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This note was uploaded on 12/10/2011 for the course STATS 371 taught by Professor Hanlon during the Fall '11 term at Wisconsin.

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371chapter1f2011 - 0 Chapter 1 Probability 1.1 Getting...

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