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Unformatted text preview: M4056 Review. October 810, 2010 Section I. Provide a precise mathematical definition for each of the following basic notions of mathematical statistics, and then provide an example or illustration: Bernoulli random variable Binomial random variable Gamma random variable Moment generating function Random sample of size n from a population with distribution f X ( x ) Sample pdf (sample pmf ) Statistic Sampling distribution of a statistic Sample mean Sample variance Normal random variable Chi squared random variable Parametric family of distributions f X ( x | ) Sufficient statistic Likelihood function Estimator Method of moments estimator Maximum likelihood estimator Mean squared error (of an estimator) Bias (of an estimator) Section II. 1. Let M be the mean of a sample of size n from a population described by a random variable X with distribution f X ( x ). Explain the difference in meaning between: the expected value of X , the sample mean (i.e., M itself), the expected value of M . 2. Let S 2 be the sample variance of a sample of size n from a population described by a random variable X with distribution f X ( x ). Explain the difference in meaning between: the variance of X , the sample variance (i.e., S 2 itself),...
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