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LECTURE NOTES on PROBABILITY and STATISTICS Eusebius …

LECTURE NOTESonPROBABILITY and STATISTICSE usebius DoedelTABLE OF CONTENTSSAMPLE SPACES1 Events5 The Algebra of Events6 Axioms of Probability9 Further Properties10 Counting Outcomes13 Permutations14 Combinations21 CONDITIONAL PROBABILITY45 Independent Events63 DISCRETE RANDOM VARIABLES71 joint distributions82 Independent random variables91 Conditional distributions97 Expectation101 Variance and Standard Deviation108 Covariance110 SPECIAL DISCRETE RANDOM VARIABLES118 The Bernoulli Random Variable118 The Binomial Random Variable120 The Poisson Random Variable130 CONTINUOUS RANDOM VARIABLES142 joint distributions150 Marginal density functions153 Independent continuous random variables158 Conditional distributions161 Expectation163 Variance169 Covariance175 Markov s inequality181 Chebyshev s inequality184 SPECIAL CONTINUOUS RANDOM VARIABLES187 The Uniform Random Variable187 The Exponential Random Variable191 The Standard Normal Random Variable196 The General Norm

Joint distributions 150 Marginal density functions 153 Independent continuous random variables 158 Conditional distributions 161 Expectation 163 Variance 169 Covariance 175 Markov’s inequality 181 ... The probability of a sequence to contain precisely two Heads is …

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