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 Normal Random Variable201 The Chi-Square Random Variable206 THE CENTRAL LIMIT THEOREM211 SAMPLE STATISTICS246 The Sample Mean252 The Sample Variance257 Estimating the Variance of a Normal Distribution266 Samples from Finite Populations274 The Sample Correlation Coefficient282 Maximum Likelihood Estimators288 Hypothesis Testing305 LEAST SQUARES APPROXIMATION335 Linear Least Squares335 Genera

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.

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  Statistics, Probability, Sets, Probability and statistics, Uncountable

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