A Course in Mathematical Statistics phần 10 pps

... Borel real line ↑, ↓ increasing (non-decreasing) and decreasing (non-increasing), respectively P, (S, A, P) probability measure (function) and probability space, respectively I A indicator of ... found in most books of linear algebra. For example, see Birkhoff and MacLane, A Survey of Modern Algebra, 3d ed., MacMillan, 1965; S. Lang, Linear Algebra, Addison-Wesley, 1968; D. C. Murdoch,...

Ngày tải lên: 23/07/2014, 16:21

... fertilizers. Comparison of a certain brand of gasoline with and without an additive by using it in several cars. Comparison of different brands of gasoline by using them in several cars. Comparison of the wearing ... distributed, under H. We may summarize all relevant information in a table (Table 1) which is known as an Analysis of Variance Table. EXAMPLE 3 For a numerical exampl...

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... Determine the minimum sample size n required by using the Binomial tables (if possible) and also by using the CLT. 13.3 .10 The number X of fatal traffic accidents in a certain city during a year may ... Testing Hypotheses 13.3.4 The rainfall X at a certain station during a year may be assumed to be a normally distributed r.v. with s.d. σ = 3 inches and unknown mean μ . For the...

Ngày tải lên: 23/07/2014, 16:21

... permutation of the integers 1 through n. From the definition of a determinant and the fact that each row and column contains exactly one 1 and the rest all 0, it follows that the n! Jacobians are ... of a normally distributed r.v. with mean a μ + b and variance a 2 σ 2 . Thus, if X is N( μ , σ 2 ), then aX + b is N (a μ + b, a 2 σ 2 ). Now it may happen that the transformation h sat...

Ngày tải lên: 23/07/2014, 16:21

... black at least 2 cards in are red exactly 1 card in is an ace the first card in is a diamond, the second is a heart and the third is a club card in is a diamond, 1 is a heart and 1 is a club 2.4.30 ... is a generic model for situations often occurring in practice. For instance, the urn and the balls may be replaced by a box containing certain items manufactured by...

Ngày tải lên: 23/07/2014, 16:21

... repeated at will, in many other cases this is, indeed, the case. In the latter case, as a rule, it is advantageous not to fix the sample size in advance, but to keep sampling and terminate the ... (as well as in the interval estimation problems to be dealt with in Chapter 15), sampling according to a stopping time is, in general, profitable, the mathematical machinery involved...

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... () ∈ [] = () ∈ [] By assuming the existence of an MLE, we have that the maximum at the left-hand side above is attained at an MLE ˆ θθ θθ θ. Then, clearly, the right-hand side attains its maximum at ˆ θθ θθ θ*, ... The Case Where Complete Sufficient Statistics Are Not Available or May Not Exist: Cramér–Rao Inequality When complete, sufficient statistics are available, the problem of findi...

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... differentiation as follows: 178 7 Stochastic Independence with Some Applications To start with, consider the probability space (S, A, P) and recall that k events A 1 , , A k are said to be independent ... a function defined on the entire real line and taking values in the complex plane. Those readers who are not well versed in matters related to complex-valued functions may feel un...

Ngày tải lên: 23/07/2014, 16:21

... improper integral in which the limits are taken symmetrically. As an example, for σ = 1, μ = 0, we have, in terms of the principal value integral, I xdx x x AA A A A A A A A * lim lim log lim ... over x, as x varies in G, of such intervals is equal to G. The same is true if we consider only those intervals corresponding to all rationals x in G. These intervals are countably...

Ngày tải lên: 23/07/2014, 16:21

... . n nj j AA →∞ = ∞ = 1 U We recall that AAAA AAA AAA AA j j ccc = ∞ =+ ∩ () +∩∩ () +⋅⋅⋅ =+ − () +− () +⋅⋅⋅ 1 11 2 1 23 121 32 U , by the assumption that A n ↑. Hence P A P A PA PA A PA A PA A PA PA A ... n n nn n nn 23 2 10 21 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +⋅⋅⋅+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =−− relationships characterizing the independence of A j , j = 1, , n and the...

Ngày tải lên: 23/07/2014, 16:21