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# Note for COMPLEX VARIABLE and STATISTICAL METHODS - CVSM By JNTU Heroes

• COMPLEX VARIABLE and STATISTICAL METHODS - CVSM
• Note
• Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP
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Classical probability “The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.” Pierre Simon Laplace (1749-1827) Pierre-Simon Laplace, A Philosophical Essay on Probabilities JNTU World Statistical Methods for Data Analysis 3

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Classical Probability Number of favorable cases Probability = Number of total cases • • Assumes all accessible cases are equally probable This analysis is rigorously valid on discrete cases only – Problems in continuous cases ( Bertrand’s paradox) P = 1/2 P = 1/6 (each dice) P = 1/4 P = 1/10 JNTU World Statistical Methods for Data Analysis 4

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What about something like this? We should move a bit further… JNTU World Statistical Methods for Data Analysis 5

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Probability and combinatorial • Complex cases are managed via combinatorial analysis • Reduce the event of interest into elementary equiprobable events • Sample space ↔ Set algebra – and/or/not ↔ intersection/union/complement E.g: 2 = {(1,1)} 3 = {(1,2), (2,1)} 4 = {(1,3), (2,2), (3,1)} 5 = {(1,4), (2,3), (3,2), (4,1)} etc. … JNTU World Statistical Methods for Data Analysis 6