Lecture8_Spring11

Lecture8_Spring11 - Elec 210: Lecture 8 Conditional...

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Elec210 Lecture 8 1 Elec 210: Lecture 8 Conditional Probability Mass Function Conditional Expected Value
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Conditional Probability Mass Function The effect of partial information about the outcome of a random experiment on the probability of a discrete random variable is reflected by the conditional probability mass function . Suppose that we know that an event C has occurred (we assume C has nonzero probability). The conditional probability mass function of X given C is By the definition of conditional probability Elec210 Lecture 8 2 (| ) [ | ] X px C P X x C == {} [] ] [ ) | ( C P C x X P C x p X = =
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Interpretation: Elec210 Lecture 8 3 S A k x k k x X = ) ( ζ ] [ ) ( k k X x X P x p = = Start by recalling unconditional pmf: Event: k A Equivalent Event ] [ k A P The pmf value is equivalent to
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Interpretation: Elec210 Lecture 8 4 S A k C x k k x X = ) ( ζ Now, with conditioning… C x X k = ) ( The conditional probability of the event “ X = x k ” is the probability of getting an outcome ζ for which X( ζ )=x k AND ζ is in C , normalized by the probability of C occurring. {} [] ] [ ) | ( C P C x X P C x p X = =
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Properties of the Conditional PMF The conditional pmf has the same properties as the pmf . The pmf is non-negative: The values of the pmf sum to 1: The conditional probability of events B defined by X can be computed by summing the conditional pmf: Elec210 Lecture 8 5 (| ) 0 X px C [ ] () in | | where XX xB P XB C p x C B S =⊂ all ) ( | ) 1 X k xS k C pxC ==  C = x S
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Example: Waiting for Bus A person arrives at a bus stop at time ζ (minutes), taking values in [1…60] with equal probability There are 12 buses, with bus n arriving at time 5 n ; i.e., the person takes bus 1 if arriving at time {1,2,3,4,5}, bus 2 if arriving at time {6,7,8,9,10}, etc.
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This note was uploaded on 03/27/2011 for the course ELEC 202 taught by Professor ? during the Spring '11 term at HKUST.

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Lecture8_Spring11 - Elec 210: Lecture 8 Conditional...

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