Lecture9_Spring11

Lecture9_Spring11 - Elec 210 Lecture 9 Important discrete...

This preview shows pages 1–8. Sign up to view the full content.

Elec210 Lecture 9 1 Elec 210: Lecture 9 Important discrete random variables Summary of variables you know: Bernoulli • Binomial • Geometric • Discrete Uniform New random variable: Poisson MATLAB commands for plotting probability mass functions and generating discrete random variables

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

View Full Document
Elec210 Lecture 9 2 Bernoulli Random Variable The Bernoulli Random Variable is simply a random variable that assumes either value 0 or 1 with probabilities (1- p ) and p . Mean and Variance Used to model Single coin toss Occurrence of an event of interest x (0) 1 (1) XX pp p p =− = [] VAR[ ] (1 ) E Xp X = 1 -p 1 p 0 p X ( x ) pmf 0 0.5 1 0 0.1 0.2 VAR( ) X p
Suppose a random experiment is repeated n independent times. For each trial, an event A occurs with probability p X, number of times that an event A occurs , is a binomial RV with Mean and Variance Applications Multiple coin flips Occurrence of a property in individuals of a population (e.g. bit errors in a transmission, defective parts in a batch) Elec210 Lecture 9 3 ( ) (1 ) for 0,1,. ... kn k X n pk p p k n k  =− =   Binomial Random Variable [] VAR[ ] ) E Xn p X np p = note that these are just n times the values for the Bernoulli (more on this later)

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

View Full Document
X = 3 Binomial Random Variable: Example PMFs Elec210 Lecture 9 4 24 0.2 n p = = 24 0.5 n p = = Symmetric for p=0.5? p = 0.8? = ?
Elec210 Lecture 9 5 Suppose a random experiment is repeated until an event A occurs. In each repeat, A occurs independently and with probability p . The number of trials until the first success, M , is a geometric RV Mean and variance: Applications Number of customers awaiting service in a queueing system Number of white dots between successive black dots in a scan of a document Number of transmissions required until an error free transmission 1 ( ) (1 ) for 1,2,. ..., k M pk p p k =− = Geometric Random Variable 2 11 [ ] VAR[ ] p EM M p p ==

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

View Full Document
Geometric RV: Example PMFs Elec210 Lecture 9 6 0.5 p = 0.3 p = X = 4 = ?
Elec210 Lecture 9 7 Let M be the number of trials until the first success in a sequence of Bernoulli trials. M is a geometric random variable. Q: What is the conditional probability that it will take k more trials until the first success, given that we have already performed m trials with no success?

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

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

{[ snackBarMessage ]}

Page1 / 24

Lecture9_Spring11 - Elec 210 Lecture 9 Important discrete...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online