By James E. Gentle
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This provides a link between exchangeability and independence. 2. A Companion for Mathematical Statistics c 2010 James E. 1 Some Important Probability Facts 15 Random Samples If the random variables X1 , . . , Xk are independent and d d X1 = · · · = Xk , we say X1 , . . d. d. random variables is called a random sample, and the cardinality of the set is called the “size” of the random sample. The common distribution of the variables is called the parent distribution of the random sample. In statistical applications we often form functions of a random sample, and use known or assumed distributions of those functions to make inferences abput the parent distribution.
D. as exponential with parameters 0, and θ, and are independent of X(i) . We have independence because the resulting joint density function factorizes. d. exponentials with parameters 0, and θ multiplied by θ is a gamma with parameters n − 2 and 1. Method of MGFs or CFs In this method, we write the MGF of Y as E(etY ) = E(eth(X) ), or we write the CF in a similar way. If we can work out the expectation (with respect to the known distribution of X, we have the MGF or CF of Y , which determines its distribution.
Now the MGF of a χ21 is ∞ 1 √ y−1/2 e−y(1−2t)/2dy 2π 0 1 = (1 − 2t)−1/2 for t < . 2 E etYi = Hence, the MGF of Y is (1 − 2t)n/2 for t < 1/2, which is seen to be the MGF of a chi-squared random variable with n degrees of freedom. This is a very important result for applications in statistics. 9 Decomposition of Random Variables It is often useful to decompose a random variable into additive components. A property of a random variable that allows a particular kind of additive decomposition is called divisibility.