By James E. Gentle
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Extra resources for A Companion for Mathematical Statistics
Now, for k = 2, . . , n, let Yk = (n − k + 1)(X(k) − X(k−1)). 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.
2. d. random variables, we only require that they be independent (and have finite first and second moments, of course). 6, we relax the hypothesis in the other direction; that is, we allow dependence in the random variables. 11 stated on page 21. 11) We are given the finite moments ν0 , ν1, . . (about any origin) of some probability distribution. Because the moments exist, the derivatives of the characteristic function ϕ(t) exist at t = 0; **** A Companion for Mathematical Statistics c 2010 James E.
37) The correlation is also called the correlation coefficient and is often written as ρX,Y . 36), we see that the correlation coefficient is in [−1, 1]. If X and Y are independent, then Cov(X, Y ) = Cor(X, Y ) = 0 (exercise). Conditional Expectations and Conditional Distributions Often the distribution of a random variable depends on the values taken on by another random variable; hence, the expected value of the random variable depends on the values taken on by the other random variable. We will use conditional expectations to develop the concept of a conditional probability distribution.
A Companion for Mathematical Statistics by James E. Gentle