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Unformatted text preview: (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 welldefined 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 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 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 = { , 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 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 } ....
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This note was uploaded on 12/15/2011 for the course STAT 5326 taught by Professor Frade during the Fall '10 term at FSU.
 Fall '10
 Frade

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