We define conditional probability as follows suppose

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We define conditional probability as follows: Suppose ( , F , P ) is a probability space and A , B F with P ( B ) > 0. Define the conditional probability of A given B as P ( A B ) P ( AB ) P ( B ) Essentially what we are saying is that the sample space is restricted from down to B . Dividing by P ( B ) provides the correct normalization for this probability measure. Some properties of conditional probability (consequences of the axioms of probability) are as follows: 1. P ( A B ) 0. 2. P ( B ) 1 3. For A 1 , A 2 , . . . F with A i A j for i j , P ( i 1 A i B ) i 1 P ( A i B ) 4. AB P ( A B ) 0 . 5. P ( B B ) 1 6. A B P ( A B ) P ( A ) 7. B A P ( A B ) 1. Definition 3 A 1 , A 2 , . . . , A n F is a partition of if A i A j for i j , and n i 1 A i , and P ( A i ) > 0. 2 Example 6 Let 1 , 2 , 3 , 4 , 5 , 6 , and A 1 1 , A 2 2 , 5 , 6 , A 3 3 , 4 . 2 The Law of Total Probability : If A 1 , . . . , A n is a partition of and A F , then P ( A ) n i 1 P ( A A i ) P ( A i ) (Draw picture). Bayes Formula is a simple formula for “turning around” the conditioning. Because conditioning is so important in engineering, Bayes formula turns out to be a tremendously important tool (even though it is very simple). We will see applications of this throughout the semester. Suppose A , B F , P ( A ) 0 and P ( B ) 0. Then P ( A B ) P ( B A ) P ( A ) P ( B ) . Why? Definition 4 The events A and B are independent if P ( AB ) P ( A ) P ( B ).
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ECE 6010: Lecture 1 – Introduction; Review of Random Variables 5 2 Is independent the same as disjoint? Note: For P ( B ) > 0, if A and B are independentthen P ( A B ) P ( A ) . (Since they are independent, B can provide no information about A , so the probability remains unchanged. If P ( B ) 0, then B is independent of A for any other event A F . (Why?). Definition 5 A 1 , . . . , A n F are independent if for each k 2 , . . . , n and each subset i 1 , . . . , i k of 1 , . . ., n , P ( k j 1 A i j ) k j 1 P ( A i j ). 2 Example 7 Take n 3. Independent if: P ( A 1 A 2 ) P ( A 1 ) P ( A 2 ) P ( A 1 A 3 ) P ( A 1 ) P ( A 3 ) P ( A 2 A 3 ) P ( A 2 ) P ( A 3 ) and P ( A 1 A 2 A 3 ) P ( A 1 ) P ( A 2 ) P ( A 3 ). 2 The next idea is important in a lot of practical problem of engineering interest. Definition 6 A 1 and A 2 are conditionally independent given B F if P ( A 1 A 2 B ) P ( A 1 B ) P ( A 2 B ) 2 (draw picture to illustrate the idea). Random variables Up to this point, the outcomes in could be anything: they could be elephants, computers, or mitochondria, since is simple expressed in terms of sets. But we frequently deal with numbers , and want to describe events associated with sets of numbers. This leads to the idea of a random variable. Definition 7 Given a probability space ( , F , P ) , a random variable is a function X mapping to R . (That is, X : R ), such that for each a R , ω : X (ω) a F . 2 A function X : R such that w : X (ω) a F , that is, such that the events involved are in F , is said to be measurable with respect to F . That is, F is divided into sufficiently small pieces that the events in it can describe all of the sets associated with X . Example 8 Let 1 , 2 , 3 , 4 , 5 , 6 , F 1 , 2 , 3 , 4 , 5 , 6 , , . Define X (ω) 0 ω odd 1 ω otherwise Then for X to be a random variable, we must have ω X (ω) a
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ECE 6010: Lecture 1 – Introduction; Review of Random Variables 6 to be an event in F .
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  • Fall '08
  • Stites,M
  • Probability, Probability theory, CDF

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