notes1 - (Probabilistic) Experiment: (, B, P ) is the...

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(Probabilistic) Experiment: (Ω , B ,P ) Ω is the sample space set of all possible outcomes. (Often denoted S.) ω denotes a particular outcome. Ω = { all possible ω } B is the class of “events” for which probabilities are defined. (We mainly ignore B in this class. Assume all events of interest have well-defined probabilities.) P is a “Probability function”. P ( A ) = probability of the event A . An event A is a subset of Ω. Experiments and events are often depicted by Venn dia- grams. 1
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Example: Roll Two Fair Dice Ω = { ( i,j ) : 1 i 6 , 1 j 6 } #(Ω) = 36 ω = ( i,j ) Example: Poker (5 card draw) Ω = set of all poker hands #(Ω) = ( 52 5 ) = 52! 5!47! a particular outcome is ω = { A , 5 , 5 ,K , 3 ♦} These are examples of experiments which are discrete, have finite Ω, have equally likely outcomes ω . In these situations: P ( A ) = #( A ) #(Ω) 2
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Example: Toss a biased coin with P (Heads) = 2 / 3 three times. Ω = { HHH, HHT, HTH, . . . , TTH, TTT } #(Ω) = 8 For ω = HTH , P ( ω ) = (2 / 3) × (1 / 3) × (2 / 3), etc. This experiment is discrete, has finite Ω, has outcomes which are not equally likely. Example: Turn on a Geiger counter for one minute and count the number of clicks. (Assume an average of λ clicks per minute.) Ω = { 0 , 1 , 2 , 3 ,... } A typical outcome might be ω = 3 . P ( ω ) is given by Poisson distribution: P ( ω ) = λ ω e - λ ω ! This experiment is discrete, has infinite (but countable) Ω, has outcomes which are not equally likely. In these situations: P ( A ) = X ω A P ( ω ) 3
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Example: Turn on a Geiger counter. Measure the length of time until you hear the first click. (Assume an average of λ clicks per minute.) Ω = (0 , ) #(Ω) = (and even worse, Ω is uncountable.) For all outcomes ω, P ( ω ) = 0 . This is an example of a continuous experiment where P is described in terms of a density function (pdf). The time has an exponential distribution and P ([ a,b ]) = Z b a λe - λx dx. P ( A ) = Z A λe - λx dx. Example: Toss a biased coin with P (Heads) = 2 / 3 infinitely many times. Record the sequence of heads and tails. Ω = { all possible sequences of H and T } . A typical ω = ( H,H,T,H,H,H,T,T,. .. ) #(Ω) = . P ( ω ) = 0 for all ω . 4
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The experiment has an infinite (and uncountable) Ω. Is this experiment discrete or continuous? How to compute probabilities P ( A )? Example: Toss a dart at a square target (1 ft. by 1 ft.). Dart is tossed “at random” (uniformly). Ω = { ( x,y ) : 0 x 1 , 0 y 1 } #(Ω) = . This is continuous experiment with P given by P ( A ) = Area( A ) Area(Ω) . Example: Now suppose the dart is tossed according to a joint density f ( x,y ) on the plane. Then (by definition) P ( A ) = Z Z A f ( x,y ) dxdy . Comment: More complicated experiments lead to higher-
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This note was uploaded on 11/16/2010 for the course STA 2023 taught by Professor Bateh during the Spring '08 term at Florida State College.

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notes1 - (Probabilistic) Experiment: (, B, P ) is the...

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