handout_week1_c - Stochastic Signals and Systems...

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Unformatted text preview: Stochastic Signals and Systems Probability Theory Virginia Tech Fall 2008 Outline 1 Conditional Probability 2 Independence of Events Conditional Probability The conditional probability of event A , given event B , is defined as P [ A | B ] = P [ A B ] P [ B ] where we assume that P [ B ] is not 0. Subjectively, one can think of the conditional probability P [ A | B ] as representing the likelihood of A s having occurred when it is known that event B has occurred. If B is known to have occurred, then A can occur only if A B occurs. Conditional Probability Frequency interpretation . Denoting by n A , n B , and n A B the number of occurrences of the events A , B , and A B , respectively, we have that P [ A ] n A n P [ B ] n B n P [ A B ] n A B n Hence P [ A | B ] = P [ A B ] P [ B ] n A B / n n B / n = n A B n B This result can be phrased as follows: If we discard all trials in which the event B did not occur and we retain only the subsequence of trials in which B has occurred, then P [ A | B ] equals the relative frequency of occurrence n A B / n B of the event A in that subsequence. Example Random experiment: A coin is tossed three times in succession. Outcomes are triplets of heads and tails....
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This note was uploaded on 05/05/2010 for the course ECECS 5605 taught by Professor Dasilver during the Fall '08 term at Virginia Tech.

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handout_week1_c - Stochastic Signals and Systems...

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