Probability and StatisticsThe revision of this well-respected text presents a balanced approach of the classical and Bayesian methods and now includes a new chapter on simulation (including Markov chain Monte Carlo and the Bootstrap), expanded coverage of residual analysis in linear models, and more examples using real data. Probability Statistics was written for a one or two semester probability and statistics course offered primarily at four-year institutions and taken mostly by sophomore and junior level students, majoring in mathematics or statistics. Calculus is a prerequisite, and a familiarity with the concepts and elementary properties of vectors and matrices is a plus. Introduction to Probability; Conditional Probability; Random Variables and Distribution; Expectation; Special Distributions; Estimation; Sampling Distributions of Estimators; Testing Hypotheses; Categorical Data and Nonparametric Methods; Linear Statistical Models; Simulation For all readers interested in probability and statistics. |
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Page 103
... random variables . This property of a marginal p.d.f. can be further illustrated by another example . Suppose that a ... Independent Random Variables It is said that two random variables X and Y are independent if , for any two sets A ...
... random variables . This property of a marginal p.d.f. can be further illustrated by another example . Suppose that a ... Independent Random Variables It is said that two random variables X and Y are independent if , for any two sets A ...
Page 119
Morris H. DeGroot. n Independent Random Variables . It is said that n random variables X1 , ... , X , are independent if , for any n sets A1 , A2 , ... , A , of real numbers , XnЄ Pr ( X1 € A1 , X2 € A2 , ... , X , = An ) Pr ( X1 € A1 ) ...
Morris H. DeGroot. n Independent Random Variables . It is said that n random variables X1 , ... , X , are independent if , for any n sets A1 , A2 , ... , A , of real numbers , XnЄ Pr ( X1 € A1 , X2 € A2 , ... , X , = An ) Pr ( X1 € A1 ) ...
Page 326
Morris H. DeGroot. 9. Suppose that six random variables X1 , ... , X6 form a random sample from a standard normal ... independent random variables , even though both are functions of the same variables X1 , ... , X. Furthermore , we shall ...
Morris H. DeGroot. 9. Suppose that six random variables X1 , ... , X6 form a random sample from a standard normal ... independent random variables , even though both are functions of the same variables X1 , ... , X. Furthermore , we shall ...
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a₁ assume B₁ Bayes estimator Bernoulli distribution beta distribution binomial distribution c₁ c₂ conditions of Exercise confidence interval Consider constant continuous distribution defined by Eq degrees of freedom Determine the value distribution with mean distribution with parameters drug Example exponential distribution following hypotheses follows from Eq form a random Furthermore gamma distribution given in Eq given value H₁ Hence joint distribution joint p.d.f. level of significance likelihood function linear loss function mean µ median null hypothesis observed values obtained P₁ Poisson distribution possible values posterior distribution Pr(X prior distribution problem random sample random variables X1 regression rejected sample mean selected at random specified standard deviation standard normal distribution statistician sufficient statistic Suppose that X1 Table tail area test procedure test the following Theorem UMP test unbiased estimator uniform distribution Var(X variance o² X₁ x² distribution Y₁ σ²