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 Random Variable206THE CENTRAL LIMIT THEOREM211SAMPLE STATISTICS246The Sample Mean252The Sample Variance257Estimating the Variance of a Normal Distribution266Samples from Finite Populations274The Sample Correlation Coefficient282Maximum Likelihood Estimators288Hypothesis Testing305LEAST SQUARES APPROXIMATION335Linear Least Squares335General Least Squares343RANDOM NUMBER GENERATION362The

S can also be uncountable, e.g., the set R of all real numbers. EXAMPLE : Record the low temperature in Montreal on January ... • (Also called sequences , vectors , ordered sets .) • The order of the items in the word is important; e.g., the word acb is different from the word bac . • The word length is the number of characters in the word.

  Statistics, Probability, Sets, Probability and statistics, Uncountable