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This course is a problem oriented introduction to the basic concepts of probability and statistics, providing a foundation for applications and further study.
At the level of Probability and Statistical Inference, Hogg and Tanis, 9th edition, Pearson
Basic Principles Covered:
- Multiplication principle, combinations, permutations
- Expected value, variance, standard deviation
- Conditional probability, Bayes rule, partitions
- Random variable, p.d.f., c.d.f., m.g.f.
- Joint distributions, marginals, conditional expectations
- Covariance, correlation
- Transformations of a random variable
- Central limit theorem, approximations
- Basic distributions: uniform, binomial, multinomial, normal, exponential, Poisson, geometric, Gamma, Chi-squared, Student t, use of tables
- Experiments, events, sets, probabilities, and random variables
- Equally likely outcomes, counting techniques
- Conditional probability, independence, Bayes' theorem
- Expected values, mean, variance, binomial and geometric distributions
- Poisson, moment generating functions
- Continuous random variables, exponential, gamma, and normal; intuitive treatment of the Poisson process and development of the relationship with the gamma distributions
- Uniform and simulation
- Multivariate distributions, calculation of probability, covariance, correlation, marginals, conditions
- Distributions of sums of random variables
- Central limit theorem
- Maximum likelihood, optimal, and unbiased estimators, examples
- Univariate transformations using the chi square as an important example
- Develop the idea of confidence intervals, confidence intervals for means with known variance in the normal case, large sample confidence intervals for means. Give the results of such a development for differences in the two-sample problem.
- Confidence intervals for variances and ratios of variances and applied problems
- Large sample confidence intervals for proportions, the one and two sample case, with examples and sample size considerations
- Introduction to formal hypothesis testing, calculation of size and evaluation of the power function. One and two sample tests of hypotheses for normal means and variances. Chi-squared goodness of fit test.