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ECSE 305 (5)


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Electrical Engineering
ECSE 305
Ioannis P.

1 Chapter 1 Introduction 1.1 Randomness versus determinism Determinism in science and engineering: ▯ Deterministic view in science: provided su▯cient information is available about the initial state and operating conditions of a natural process or a man-made system, its future behavior can be predicted exactly. ▯ This operational viewpoint has been the prevailing one in most of your college and university education (mechanics, circuit theory, etc.) ▯ A typical example is provided by classical mechanics: - Consider the motion of a particle under the in uence of various forces in three-dimensional space. - If we know the initial position and velocity vectors of the particle, its mass and the total force ▯eld, Newton’s laws can be used to calculate (i.e. predict) the future trajectory of the particle. 2003 Ben▯t Champagne Compiled January 11, 2012 1.1 Randomness versus determinism 2 The concept of randomness: ▯ The above view is highly idealistic: In most "real-life" scienti▯c and engineering problems, as well as many other situations of interest (e.g. games of chance), we cannot do exact predictions about the phenomena or systems under consideration. ▯ Two basic reasons for this may be identi▯ed: - we do not have su▯cient knowledge of the initial state of the system or the operating conditions (e.g. motion of electrons in a micropro- cessor circuit). - due to fundamental physical limitations, it is impossible to make exact predictions (e.g. uncertainty principle in quantum physics) ▯ We refer to such phenomena or systems as random, in the sense that there is uncertainty about their future behavior: a particular result or situation may or may not occur. ▯ The observation of speci▯c quantities derived from such a random system or phenomenon is often referred to as a random experiment. 2003 Beno^▯t Champagne Compiled January 11, 2012 1.1 Randomness versus determinism 3 Examples: ▯ Consider the following game of chance: - We roll an ordinary six-sided die once and observe the number show- ing up, also called outcome. - Possible outcomes are represented by the set of numbers S = f1;2;3; 4;5;6g. - We cannot predict what number will show up as a result of this experiment. - Neither can we predict that a related event A, such as obtaining an even number (represented by A = f2;4;6g), will occur. ▯ Consider a more sophisticated example from communications engineer- ing: - Using an appropriate modulation scheme, we transmit an analog speech signal s(t) over a radio channel. - Due to channel and receiver noise, and other possible disturbances during the transmission, the received signal r(t) is generally di▯erent from the transmitted one, i.e. s(t). - In general, it is not possible for the radio engineer to predict the exact shape of the error signal n(t) = r(t) ▯ s(t). 2003 Beno^▯t Champagne Compiled January 11, 2012 1.2 The object of probability 4 1.2 The object of probability Regularity in randomness: ▯ OK, we cannot predict with certainty the particular outcome in a single realization of a random experiment, but... ▯ In many practical situations of interest (games of chance, digital commu- nications, etc..), it has been observed that when a random experiment is repeated a large number of times, the sequence of results so obtained shows a high degree of regularity. ▯ Let us be more speci▯c: Suppose we repeat a random experiment (e.g. rolling a die) n times. Let ▯(A;n) be the number of times that a certain event A occur (e.g. the result is even). It has been observed that ▯(A;n) ! constant as n ! 1 (1.1) n ▯ The ratio ▯(A;n)=n is called the relative frequency. ▯ The constant provides a quantitative measure of the likelihood of A. 2003 Beno▯t Champagne Compiled January 11, 2012 1.2 The object of probability 5 Example: ▯ Consider a simple experiment consisting in ipping a coin. Let A denote the event that a head shows up. ▯ Now suppose this experiment is repeated n times. The quantity ▯(A;n) simply represents the number of times that head shows up, out of n similar ips of the coin. ▯ Assuming that the coin is fair (unloaded), we expect the ratio ▯(A;n)=n to approach 1/2 as n gets larger and larger. ▯ Results of a computer simulation experiment: - Sequence of observed outcomes: THTHHH... - Relative frequency versus n: 1 0.9 0.8 0.7 0.6 0.5 relative frequency 0.3 0.2 0.1 100 10 102 10 number of repetitions - Example:
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