Chapter 04 Problems - Exercises probabilities of each of...

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Unformatted text preview: Exercises probabilities of each of th'n' by summation Hail the putt? ‘ EXERCISES E4.1 E4..2 E43 E44 E45 if“ ={w1.w2.w3}.- A =29,s1n_d P({m1,w2}) ..... 65-P({w2'1 @3D— "' “85 then determine the pmf {pi} and P({to;, a13}).. - In a certain communication system a byte (8bits) is transmitted with a bit error probability of .1. If the system can correct at most one error made in eachbyte, what is the probability of a message being received correctly? What is the most probable number of errors in a byte? If the most probable number of errors in a byte IS 2 what can you say about the probability of no errors? Use the binomial pmf to evaluate the probability that, in a message of length 5, there will be fewer than two errors when the probability of an erroron a single symbol' is 0 1. . If a byte is as likely to contain exactly three errors as it is to contain exactly four errors, what is the probability of its containing exactly. three errors? E43 E43 new I £14.11 1754.12 1314.13 E4.l4 _____ .___-_-_.__...._._.________. ______ Chapter-4 Describing Probability I: Countable 52 If all bytes are equally likely to be received, evaluate the probability of receiving a byte having at least four more ones than zeros. In many samples of text of' length 8 characters, it is observed that the most probable I'number of errors is 2. _ ' a. What probability model do you suggest for the number X of errors? b. Is a per symbol error probability of U3 compatible with the observed most probable ,- number of errors being two? c. If the per symbol error probability is l/3, then what is the probability that the - number of errors is two? In a_ certain bit transmission system in which either a “0” or a “1” is sent, it is observed that the probability of making two errors in 10 bits is .0045 What' rs the probability of making an error on the first bit? _ In the transmission of a packet of 100 bits that is received with possible errors, the probability of there being exactly one error in the packet is twice the probability of I there being exactly two errors. Selecting an appropriate probability model and making reasonable approximations, what is the average number of errors? A packet of length 100 bits is transmitted over a noisy channel with error correction that can correct no more than two errors. The probability that the id: bit will be in ‘. error is .01_._ a. What rs the probability pc that the packet, after possible error correction, will be received correctly? b. What rs the most probable number of errors9 A communications channel transmits a block of S symbols in time T , with the probability . .p of a given symbol being in error. An error correction system allows'the receiver to correct up to k errors, and if more than 1: errors are made over the channel, then the block" rs incorrectly decoded by the receiver. What is the probability that the block will becorrectlydecodedifS -5- k=1,p= .0]? A sequence of 10 unlinked (independent) bytes has been received. It is known that the probability is .3 that a first symbol ina byte is‘ a 0.. Let K be the number of received bytes having a 0 as first symbol. What is P0? = 2)? In a certain communication system, the probability is .1 that a packet will arrive that is in err0'r. ' a What' rs the probability of no errors in the next 10 packets received? b. What is the most probable number of errors made in the 10 packets and how probable is it? _ - - , In a long sequence of it calls to Computer memory, we find that calls to cache and calls to main memory are made so that the probability of any particular sequence of these It calls involving exactly c calls to the cache rs given by p‘ (1 p)" "‘ ,for known 0 < p < 1. a. What' is. the probability of there being exactly c calls to the 'cache without regard to the sequencing of the calls? b. What' rs the most probable number of calls to cache? OQ1 , Exercises E415 E416 E447 £14.18 1514.19 1134.20 114.21 ' E422 E433 153 c. Show that if (:1 < 111 < n, then the probability of or calls to cache in the first 111 calls to memory is (2 Ip) l“(1,11)"1 ‘1. d. What rs the probability that the first 1 calls are only to‘ the cache? e. What is the probability of the rth call being the first call to main memory? A component has a random lifetime L, measured in units of'days, such that the pinbability P(L = n) of failure on day 11 is given by the geometric distribution 9(5), P(L=n)I=(l—-fi)fi.", 11 EN' If [3- —— i, then calculate the probability of the event F- of no failure before day 1, F,- ={1‘, i+1, i+2,. }..={n:nzi_}. (11611111111111 0 < b < 1, 2,31,, 1'": -b"/(l -- 11).) . If the mean or average lifetime L of a component is 10 days, then what is an appropriate probability model for L‘? A device that fails without aging effects is known to have a mean lifetime of 3 hours. What' rs the probability that it will last either less than 1 or more than 5 hours? What is the pmf for the trial number T to the first occurrence Iof a “3” in repeated tosses of a fair die? 1 Let T: '1, 2. :be the trial number on which a 5- or a 6 first appears in successive tosses of a fair the What is IP.(T= k)? In a particular implantation process the average number of ions that are implanted in a small surface is 100 The device works successfully if the number of ions actually implanted rs in the range [80,120]. Provide an expression for the probability P(S) that - a sample device will work successfully. at. What is the probability that a laser emitting an average of y photonslsec will be undetectable because it emits no photons' 1n time T? . b. What' 1s the most probable number of photons emitted by this source in time T and how probable 1s it? If the average number of clients requesting service from a file server in one minute is 3, what' 15 the probability of no clients requesting service in one minute? An optical communications receiver is confronted With a light source such that, over ‘ a Lasso interval, we are twice as likely to observe 3 photons as we are to observe 4 - photons. 4 , - E424 E435 E426 a. What' 1s the most likely number of photons to be observed 1n 1 pseco? b “What is the probability that we will observe any photons in a 1-11st interval (detect the presence of the light source)? If the probability of 0 photons being emitted' in time T rs. .1, Ithen what is the probability of at least 2 photons being emitted in I. ‘7 The probability of observing 2 photons' m a given time is ItWice the probability of observing 3 photons. What rs the most probable number of photons to be Iobserved? If a Geiger counter measures an average rate of 2» decayslseco, then what is the proba- bility of its measuring no decays in a given time interval of length l seco‘? What is the most probable number of particles to be observed? - * 154 E427 £4.28 E429 E430 134.31 134.32 134.33 I 13434 E435 E436 . E437 Chapter 4 Describing Probability I: Countable $2 The number'KT of ions emitted from a source in time T satisfies P(K1- 2 1) = ..l a. What is the mean or average number of ions emitted in ”I ? b. :What is the probability P(K21r = 2.) of 2 ions arriving in time 27"? , In a randomly generated binary sequence of length m, the probability of a run of zeros of exactlength r(< m -'- 1), starting at position _k, is p”(l p)2' if 1 < It < m — r', the prob- ability is p"(1 -— p) if k = 1, m r-- r and is zero if k > m - 1'. Use Boole’s Inequality to uppers-bound the probability that the binary sequence will contain a run of exact length r. The number N of voice communication packets arriving in time T is such that P(N = 0) a P(N = 1).. What is the probability description ofN ? If the probability is .9 of observing at least one emitted photon in 1 second, what is the probability of observing exactly one photon in 1 second? What is the most probable number of photons to be observed 1n 1 second? We know that the probability rs 6'40 of no packets being received by a client computer in 1 second. _ a- What rs the probability law for the random variable N of number of packets received in 1 second? b. What 1s the probability of no more than two packets being received 1n 1 second? -..c. What' is the most probable number n* of packets to be received' in 2 seconds and how probable' rs it? :The number X of 10118 implanted' 1n 1 second' 1n 2 small area a of a device is such that "'P(X= 2)= 3P(X= 3). ' a. Fully specify the probability mass function px. b. What 1s the probability of at least three inns being implanted? Specify the pmf needed to model the following random sources: a. number N of clicks in a Geiger counter in _5 seconds when the average number of ‘ clicks in 1 second is 3; _ b. number N of errors made 1n transmitting a text of 100 characters when the average _ number of errors in such a text is 1; The probability of five customers arriving in a queue in 2 minutes is twice as great as the probability of two customers arriving in 2 minutes. What is the, probability of no customers arriving in 5 ntinutés? What' rs the probability that at least two packets will be received" 1n 3 milliseconds when the average number of packets received to l millisecond' is two? a. Numericaily evaluate p1 for 3(3/2) b. What" 15 p2? The sizes of files stored on a large Unix file system follows a 2(a) pmf, when file sizes are measured in units of kilobytes a. If file sizes of 1 KB are 10,000 more probable than file sizes of 1 MB, then what is the parameter or? b. How much more probable are file sizes of 1 MB than those of 1 GB? Exercises 1 55 134.38 If the probability p] that a file requested on the Web will have size 1 KB is 10 times as large as the probability 192 that the file size is 2 KB, then compare p; with the probability _ p3 that the file size is 8 KB E439 The random number N of repeated, independent experiments 8;, 1., SN that must be performed until we first observe the rth occurrence of a specified event A of probability PM) a p is described by the negative binomial pmf given by PcN =n5=pi"’ = ("‘1 r _1)P’(1-p)”", n = r,r+ 1,..r. This pmf can be understood by realizing that any specific sequence of r occurrences of A and n r occurrences of A“ has probability p’ (1 - p)""" However, fin is to be the time of occurrence of the rth appearance of A, then there had to be r_ -— l occurrences in the preceding it -—— 1 trials and the rth occurrence of‘A precisely on the nth experimental _ outcome. There are ”3) sequences with this property. ' a. If r = 1, relate this to the Geometric pmf' 9(5)“ . be. For a given p and r, what is the most probable value of N? E4.40 a. What is the probability that a fourth “head” will appear on the eighth toss of a fair coin? . . _ by What is the probability if the coin is biased with probability 1/3 for “head”? ...
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