Internet Protocols CSC 343
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This 10 page Class Notes was uploaded by Melyssa Aufderhar on Wednesday October 28, 2015. The Class Notes belongs to CSC 343 at Wake Forest University taught by Errin Fulp in Fall. Since its upload, it has received 13 views. For similar materials see /class/230722/csc-343-wake-forest-university in ComputerScienence at Wake Forest University.
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Date Created: 10/28/15
Probability Review csc 343643 WAKE FOREST Department of Computer Science Fall 2008 lnternet Protocols Probability Review Basic Definitions Random experiment Observing the outcome of a chance event Elementary outcomes All possible results of the experiment Sample space Set of all the elementary outcomes Consider tossing a coin Random experiment consists of recording the outcome Elementary outcomes are heads and tails Sample space is headstais What about rolling a single die E w Fulp Fall 2008 1 m lnternet Protocols Probability Review 2 Probability 0 We want to assign a numerical weight to each outcome This probability measures the likelihood of it occurring Denoted as which is the probability of xi 0 Consider rolling a fair die and recording the side facing up Since the die is fair all sides have an equal probability M1 M2 M3 p4 M5 M6 0 Of course a loaded die would have one side appear more often M1 i M2 M3 M4 M5 M6 What is the assumption for the remaining sides E W Fulp Fall 2008 lnternet Protocols Probability Review 3 Characteristic Properties of Probability o Probabilities are never negative 2 0 Probability of zero means the event cannot happen 0 Probabilities are never greater than 1 ME S 1 Probability of 1 means the event is certain to happen 0 The total probability of all possible outcomes is one px1 pxn 1 13971 Did the previous ezramples adhere to these rules w Fulp Fall 2008 lnternet Protocols Probability Review 4 Events 0 An event is a set of elementary outcomes The probability of an event is the sum of the probabilities of the elementary outcomes 0 Consider rolling a pair of dice Event Elementary Outcomes Probability A dice add to three 1221 pA g7 B dice add to siX 1524334251 pB gr Cfirst die showsl 111213141516 pC e D second die shows 1 112131415161 pD gr Where did the denominator 0f 36 come from m w Fulp Fall 2008 lnternet Protocols Probability Review 5 Combining Events 0 Given two events we can combine them to create a new event E and F the event E and the event F both occur E or F the event E or the event F or both occur not E the event E does not occur 0 Remember events can be represented as sets Combining events is really just set manipulation E W Fulp Fall 2008 lnternet Protocols CSC 39 36 I Probability Review 6 Addition Rule 0 For any events E and F ME or F pE pF imE and F Adding double counts the elementary outcomes shared by E and F so we must subtract the extra amount 0 What is the probability of the first or second die being one 6 1 pC or D pC PD PJEC and D 73 36 36 7 o If E and F are mutually exclusive then the two events will not double count as a result ME or F ME PF What is a dice ezmmple of this E w Fulp lnternet Protocols CSC 39 36 I Fall 2008 Probability Review 7 Multiplication Rule o For any events E and F ME and F PE1FPF Where pEF is the conditional probability of E will occur given that F has already occurred If E and F are independent then the occurrence of one has no influence on the other ME and F PE PF o What is the probability of snake eyes7 pboth die are 1 pfirst die is 1 psecond die is 1 1 1 1 86736 E w Fulp Fall 2008 m lnternet Protocols C lnternet Protocols C Probability Review 8 Subtraction Rule 0 For any event E pnot E 17E o What is the probability of not rolling snake eyes pboth die are not 1 1 ipboth die are 1 1735 36736 m W Fulp Fall 2008 Probability Review 9 Random Variables o A random variable X is the numerical outcome of an experiment 0 For example X could represent The sum of the faces of a pair of dice Number of heads when two coins are tossed Number of bits transmitted in error in a frame 0 We want to observe the probabilities of the outcomes The probability of the RV X having the value x PrX x pm W Fulp Fall 2008 lnternet Protocols C Probability Review 10 m RV Examples 0 Consider tossing two coins let X number of heads a How did we get these probability values 0 Consider rolling a pair of dice let X sum of the dice W o The mean of a random variable X is Vac Sum of the possible values each weighted by its probability o Is also called the expected value of X eX o The expected number of heads when tossing two coins is 2 1 1 1 6leinpxi011 2Z1 i0 m 11 i z l2l3l4l5l6l7l8l9l10lllll2l 1 2 3 4 5 6 5 4 3 2 1 lprlXllllEl l l lElElll Fuip FaiiQOOS lnternet Protocols C Probability Review Mean of a RV w Fuip Faii 2008 lnternet Protocols Probablllty Revlew 12 Bernoulli Trial 0 An experiment trial that has the following properties 1 Result of a trial is either a success or a failure 2 The probability p of success is the same in every trial 3 Independence one trial has no influence on later outcomes 0 Often used in network modeling What are some other ezrample experiments that can be considered a Bernoulli trial E W Fulp Fall 2008 lnternet Protocols Probablllty Revlew 13 Binomial RV o X is the number of successes in n repeated Bernoulli trials with probability p o For example number of heads in two flips of a coin 71 25p l k number of successes l 0 l 1 l 2 l l who lililil How did we get the probability values E W Fulp Fall 2008 m lnternet Protocols m lnternet Protocols Probability Review 1 o The equation is P7 X k W1 710M 0 Where n 7 nl k i kln 7 kl ls called n choose k which counts the possible ways of getting k successes in n trials W Fulp Probability Review 1 Fall 2008 m Example problem Three bits are grouped together to form a frame and transmitted over a channel Assuming the probability of a single bit error is 025 and errors are independent 0 The elementary outcomes are Let b represent the orginal bit e represent an error Bit Pattern Probability bbb r 0421875 ebb 0140625 beb 0140625 bbe 0140625 eeb r 0046875 bee r r 0046875 ebe r r r 0046875 eee r r r 0015625 W Fulp Fall 2008 lnternel Protocols Probablllty Revlew o What is the probability that a frame has at least one error Could use brute force pat least one error p ebb p beb p bbe peeb p bee p ebe p eee 0578125 Could use Binomial RV pat least one error p1 error p2 errors p3 errors 1 2 2 1 3 0 iltgtlt2gtlt3gtlt1gtlt2gtlt2gtlt1gtlt2gt 0578125 Could use not pat least one error 1 7 pno error 1 7 01421875 01578125 m w Pulp Fall 2008 lnternet Protocols Probablllty Revlew 1 u 0 Assume the third bit is a parity bit What is the probability of an undetected error Parity cannot detect an even number of errors Brute force p2 errors p eeb p bee p ebe 01140625 Binomial RV lt j gt 32 31 0140625 E w Pulp Fall 2008 o lnternet Protocol Probablllty Rewew m o What is the average number of transmissions required to receive the frame With no errors Let g perror in frame 17 g pno error in frame and i be a RV that represents the number of transmissions If i transmissions are required then we would have 7 1 failures followed by 1 success Use the expected RV formula 00 co Zip transmissions 9171 1 7 g i0 i0 Eli 1 1 7 37 1 7 g 1 7 01578125 18 W Fulp Fall 2008
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