MIT2_017JF09_ch03

MIT2_017JF09_ch03 - 3 PROBABILITY 3 15 PROBABILITY In this...

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3 PROBABILITY 15 3 PROBABILITY In this section, we discuss elements of probability, as a prerequisite for studying random processes. 3.1 Events Define an event space S that has in it a number of events A i . If the set of possible events A i covers the space completely, then we will always get one of the events when we take a sample. On the other hand, if some of the space S is not covered with an A i then it is possible that a sample is not classified as any of the events A i .Ev en t s A i may be overlapping in the event space, in which case they are composite events; a sample may invoke multiple events. But the A i may not overlap, in which case they are simple events, and a sample brings only one event A i , or none if the space S is not covered. In the drawing below, simple events cover the space on the left, and composite events cover the space on the right. A 2 A 3 A 1 A 4 A 5 A 2 A 1 A 3 A 5 A 4 S S Intuitively, the probability of an event is the fraction of the number of positive outcomes to the total number of outcomes. Assign to each event a probability, so that we have p i = p ( A i ) 0 p ( S )=1 . That is, each defined event A i has a probability of occurring that is greater than zero, and the probability of getting a sample from the entire event space is one. Hence, the probability has the interpretation of the area of the event A i . It follows that the probability of A i is exactly one minus the probability of A i not occuring: p ( A i p ( A ¯ i ) . Furthermore, we say that if A i and A j are non-overlapping, then the probability of either A i or A j occuring is the same as the sum of the separate probabilities: p ( A i A j )= p ( A i )+ p ( A j ) .
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M A i 3 PROBABILITY 16 Similarly if the A i and A j do overlap, then the probability of either or both occurring is the sum of the separate probabilities minus the sum of both occurring: p ( A i A j )= p ( A i )+ p ( A j ) p ( A i A j ) . As a tangible example, consider a six-sided die. Here there are six events A 1 ,A 2 3 4 5 6 , corresponding with the six possible values that occur in a sample, and p ( A i )=1 / 6 for all i . The event that the sample is an even number is M = A 2 A 4 A 6, and this is a composite event. 3.2 Conditional Probability If a composite event M is known to have occurred, a question arises as to the probability that one of the constituent simple events A i occurred. This is written as P ( A j | M ), read as ”the probability of A j , given M ,” and this is a conditional probability. The key concept here is that M replaces S as the event space, so that p ( M ) = 1. This will have the natural effect of inflating the probabilities of events that are part of event M , and in fact p ( A j | M p ( A j M ) . p ( M ) Referring to our die example above, if M is the event of an even result, then we have M = A 2 A 4 A 6 p ( M A 2 p ( A 2 / 6 p ( M / 2 −→ 1 / 6 p ( A 2 | M 1 / 2 =1 / 3 .
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MIT2_017JF09_ch03 - 3 PROBABILITY 3 15 PROBABILITY In this...

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