Lecture8_Spring11

# Lecture8_Spring11 - Elec 210: Lecture 8 Conditional...

This preview shows pages 1–7. Sign up to view the full content.

Elec210 Lecture 8 1 Elec 210: Lecture 8 Conditional Probability Mass Function Conditional Expected Value

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 = =
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 = =
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/27/2011 for the course ELEC 202 taught by Professor ? during the Spring '11 term at HKUST.

### Page1 / 18

Lecture8_Spring11 - Elec 210: Lecture 8 Conditional...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online