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Unformatted text preview: Review of Probability Theory Probability Define a probability space S as a set comprising all outcomes i ξ of an experiment. Define an event A as a set of i ξ . So, A S i . We can use the Set Theory to explain the theory of probability. Assign to each event A a real number ( ) P A . We call it the probability of event A . Probability must satisfy the following three axioms: 1. ( ) P A e 2. ( ) 1 P S = 3. If { } AB φ = , then ( ) ( ) ( ) P A B P A P B + = + Countable spaces: If S is composed of N outcomes, we call it a countable space. Here N is a finite number. In a countable space, we can use ( ) i i P P ξ = to find the probability for any event. If S is composed of uncountable infinity of elements, we can not use the probabilities of its outcomes to calculate the probability of an event. We must use other method to define probability. The conditional probability of event A given event U can be defined as ( ) (  ) ( ) P AU P A U P U = where ( ) P U i . Total probability: The probability of B can be calculated by the combination of its conditional probabilities given events i A from a partition of S , i.e., 1 1 ( ) (  ) ( ) (  ) ( ) n n P B P B A P A P B A P A = + + L where the event set { } 1 n A A L form a partition of S . Note: for and i j i i A A i j A S φ = = U Bayes’ Theorem: (  ) ( ) (  ) ( ) i i i P B A P A P A B P B = Events A and B are independent, if ( ) ( ) ( ) P AB P A P B = The Cartesian Product 1 2 S S S = of two experiment spaces form a space S . Its events are formed by the Cartesian Product A B i of two events 1 2 and A S B S . If 1 2 and S S are independent experiments, then ( ) ( ) ( ) P A B P A P B i = for any 1 2 , A S B S . Bernoulli trials: If a set consists of n elements, then the number of its subsets comprising k elements is ! !( )! n n k k n k = . If the probability of head is ( ) P h p = when we toss a coin, then the probability that k times of head occur when we toss the coin n times is ( ) k n k n n P k p q k = , where 1 q p =  . If the probability of event A in an experiment space is p , then the probability that event A occurs k times when we repeat the experiment n times is the same as that shown above. Random Variable (RV) If we map all experimental outcomes i ξ of a probability space S to real values i x R i , then { } x i x is an event. The event is the set of all outcomes i ξ that are assigned to real values less than x . Similarly, we can define the event 1 2 { } x x i x as the set of all outcomes i ξ that are assigned to real values in the interval 1 2 [ , ] x x . A random variable x is defined by assigning a number ( ) x ξ to every outcome ξ of S . By this definition, { } x i x is an event for any x , and we have ( ) 1 and ( ) P x P x < = =  C = In other words, events which are originally defined as subsets of S are now defined on R ....
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This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sinhorngchen during the Fall '08 term at National Chiao Tung University.
 Fall '08
 SinHorngChen

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