lecture04

# lecture04 - ECE 5510 Random Processes Lecture Notes Fall...

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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 4 Today: (1) Combinations and Permutations, (2) Discrete ran- dom variables Announcements: • Discussion item schedule is up. • Homework 1 due today at 5pm, solutions will be posted to- morrow just after 5pm. • OH today are 1:00-3:00. • Homework 2 is posted online, it is a little tougher than HW 1. • Today’s reading: 1.8, 1.9, 2.1-2.4. For Tuesday, read: 2.5, 2.7, 2.8 1 Combinations Example: What is the probability that two people in this room will have the same birthday? Assume that each day of the year is equally likely, and that each person’s birthday is independent. 1. How many ways are there for n people to have their birthday? Answer: Each one is chosen independently, assume 365 days per year. So: 365 n . 2. How many ways are there to have all n people have unique birthdays? The first one can happen in 365 ways, the second has 364 left, and so on: 365 P n = 365! / (365 − n )!. 3. Discrete uniform probability law: P [ ∃ no duplicate birthdays] = 365! / (365 − n )! 365 n 4. See Fig. 1. ECE 5510 Fall 2009 2 10 20 30 0.25 0.5 0.75 1 Pr[No Repeat Birthdays] Number of People Figure 1: Tree. Example: Poker (five card stud) Evaluate the probabilities of being dealt poker hands. The standard deck has 52 cards, 13 cards of each suit; and there are four suits (hearts, diamonds, clubs, and spades). The thirteen cards are, in order, A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. The ace (A) can also be higher than the king (K). 1. How many different hands are there? A: 52 choose 5, or 2,598,960. 2. P [ StraightFlush ]? (a straight flush consists of five cards of the same suit, in a row: (4,5,6,7,8), all hearts, for example.) A: Starting at the (A, 2, 3, 4, 5) hand through the (10, J, Q, K, A) hand, there are 10 straight flushes in each suit. So P [ straightflush ] = 40 2 , 598 , 960 ≈ 1 . 5 × 10 − 5 3. P [ Flush ]? (A flush is any 5 cards of the same suit, not in- cluding any straight flushes.) There are 13 of each suit, so 13cluding any straight flushes....
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lecture04 - ECE 5510 Random Processes Lecture Notes Fall...

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