LECTURE NOTES on PROBABILITY and STATISTICS Eusebius …

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LECTURE NOTESonPROBABILITY and STATISTICSEusebius DoedelTABLE OF CONTENTSSAMPLE SPACES1Events5The Algebra of Events6Axioms of Probability9Further Properties10Counting Outcomes13Permutations14Combinations21CO NDITIONAL PROBABILITY45Independent Events63DISCRETE RANDOM VARIABLES71Joint distributions82Independent random variables91Conditional distributions97Expectation101Variance and Standard Deviation108Covariance110SPECIAL DISCRETE RANDOM VARIABLES118The Bernoulli Random Variable118The Binomial Random Variable120The Poisson Random Variable130CONTINUOUS RANDOM VARIABLES142Joint distributions150Marginal density functions153Independent continuous random variables158Conditional distributions161Expectation163Variance16 9Covariance175Markov s inequality181Chebyshev s inequality184SPECIAL CONTINUOUS RANDOM VARIABLES187The Uniform Random Variable187The Exponential Random Variable191The Standard Normal Random Variable196The General Normal Random Variable201The Chi-Square

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 …

  Statistics, Joint, Probability, Density, Probability and statistics