An Introduction to Probability Theory and Its Applications, Volume 1A complete guide to the theory and practical applications of probability theory An Introduction to Probability Theory and Its Applications uniquely blends a comprehensive overview of probability theory with the real-world application of that theory. Beginning with the background and very nature of probability theory, the book then proceeds through sample spaces, combinatorial analysis, fluctuations in coin tossing and random walks, the combination of events, types of distributions, Markov chains, stochastic processes, and more. The book's comprehensive approach provides a complete view of theory along with enlightening examples along the way. |
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Page 217
... ( 1.1 ) . If two random variables X and Y are defined on the same sample space , their joint distribution is given by ( 1.3 ) and assigns probabilities to all combinations ( x , y ) of values assumed by X IX.1 ] 217 RANDOM VARIABLES.
... ( 1.1 ) . If two random variables X and Y are defined on the same sample space , their joint distribution is given by ( 1.3 ) and assigns probabilities to all combinations ( x , y ) of values assumed by X IX.1 ] 217 RANDOM VARIABLES.
Page 218
... independent if , for any combination of values ( x , y , ... , w ) assumed ... independent trials . Comparing this definition to ( 1.13 ) , we see that if Xx depends only on the outcome of the kth trial ... RANDOM VARIABLES ; EXPECTATION.
... independent if , for any combination of values ( x , y , ... , w ) assumed ... independent trials . Comparing this definition to ( 1.13 ) , we see that if Xx depends only on the outcome of the kth trial ... RANDOM VARIABLES ; EXPECTATION.
Page 242
... independent random variables with a common distribution ; let its mean be m , its variance o2 . Let X = ( X1 + ··· + Xn ) / n . Prove that16 = = 1 n n k = 1 — E ( 2 , ( x , x ) ) = ( X- -X ) 2 = σ2 . Prove that 39. Let X1 , ... , X , be ...
... independent random variables with a common distribution ; let its mean be m , its variance o2 . Let X = ( X1 + ··· + Xn ) / n . Prove that16 = = 1 n n k = 1 — E ( 2 , ( x , x ) ) = ( X- -X ) 2 = σ2 . Prove that 39. Let X1 , ... , X , be ...
Contents
CHAPTER PAGE | 1 |
THE SAMPLE SPACE | 7 |
ELEMENTS OF COMBINATORIAL ANALYSIS | 26 |
Copyright | |
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a₁ applies arbitrary assume balls Bernoulli trials binomial coefficient binomial distribution cards cells central limit theorem chance fluctuations chapter coin conditional probability consider corresponding defined denote dice digits elements equally probable event example expected number experiment Find the probability finite follows frequencies function genes genotypes geometric distribution given hence inequality infinite integer intuitive joint distribution k₁ large numbers law of large lemma limit theorem means mutually independent n₁ negative binomial distribution normal approximation Np(k nth trial number of paths number of successes observed outcomes P₁ pairs pairwise independent particles path of length player Poisson distribution population possible probability distribution probability theory problem proof Prove r₁ random walk represents result S₁ S₂ sample points sample space statistics Stirling's formula stochastically independent Suppose tossing total number values variance X₁