Mathematical Statistics

Ngày tải lên: 08/04/2014, 12:29

... which are subsets of A measurable space k-dimensional Euclidean space and Borel σ-field, respectively Borel real line increasing (non-decreasing) and decreasing (non-increasing), respectively probability ... r1, r2 d.f and noncentrality parameter δ } } expectation (mean value, mean) of X variance (standard deviation) of X Covariance and correlation coefficient, respectively, of X and Y n char...

Ngày tải lên: 23/07/2014, 16:21

... vertical (straight) lines In such a case there are formed IJ rectangles in the resulting rectangular array which are also referred to as cells (see also Fig 17.1) The same interpretation and terminology ... under H We may summarize all relevant information in a table (Table 1) which is known as an Analysis of Variance Table REMARK EXAMPLE For a numerical example, take I = 3, J = a...

Ngày tải lên: 23/07/2014, 16:21

... there are gender-associated differences in mathematical ability (as is often claimed!) Take α = 0.05 Boys: 80 96 98 87 75 83 70 92 97 82 Girls: 82 90 84 70 80 97 76 90 88 86 (Hint: Group the grades ... testing H against A, consider the following sequential procedure: As long as a < λn < b, take another observation, and as soon as λn ≤ a, stop sampling and accept H and as soon...

Ngày tải lên: 23/07/2014, 16:21

... parameter of interest, the remaining parameters serving as nuisance parameters Writing down the joint p.d.f of the X’s and Y’s and reparametrizing the family along the lines suggested in Remark ... Testing Certain Composite Hypotheses 3 57 13.5.4 A manufacturer claims that packages of certain goods contain 18 ounces In order to check his claim, 100 packages are chosen at random from a...

Ngày tải lên: 23/07/2014, 16:21

... Remark 1, it can be seen that the Cramér– Rao bound is again equal to 2θ 2/n As a matter of fact, we arrive at the same conclusion by treating θ1 as a constant and θ2 as the (unknown) parameter ... justifiable by a purely mathematical reasoning, it does provide a method for producing estimates in many cases of practical importance In addition, an MLE is often shown to have severa...

Ngày tải lên: 23/07/2014, 16:21

... variable in the integral on the right-hand side above applies (see for example, T M Apostol, 216 Transformations of Random Variables and Random Vectors Mathematical Analysis, Addison-Wesley, 1 957 , ... usual Euclidean distance in ‫ ޒ‬k.) 8.6* Pólya’s Lemma and Alternative Proof of the WLLN The following lemma is an analytical result of interest in its own right It was used in the c...

Ngày tải lên: 23/07/2014, 16:21

... derives its name from the property just established As has already been seen in the first two examples in Section of Chapter 5, factorial moments are especially valuable in calculating the variance of ... with parameters r and p 7 .4* 177 Independence of Classes of Events and Related Results 7.1 Stochastic Independence: Criteria of Independence 7.3.3 The life of a certain part in a...

Ngày tải lên: 23/07/2014, 16:21

... we have a theorem strictly analogous to Theorems and and also remarks such as Remark 1(i)–(iv) following Theorem In particular, the analog of (iv) says that F(x1, , xk) has kth order partial ... preliminary concepts and results are needed and will be also discussed DEFINITION A set G in ‫ ޒ‬is called open if for every x in G there exists an open interval containing x and contained...

Ngày tải lên: 23/07/2014, 16:21

... Britannica are arranged on a shelf What is the probability that: i) All 24 volumes appear in ascending order? ii) All 24 volumes appear in ascending order, given that volumes 14 and 15 appeared in ... ; at least cards in s are red}, = {s ∈ S ; exactly card in s is an ace}, = {s ∈ S ; the first card in s is a diamond, A1 = s ∈ S ; all cards in s are black , A2 A3 A4 } th...

Ngày tải lên: 23/07/2014, 16:21

... Also A1 ∪ (A2 ∪ A3 ) = (A1 ∪ A2 ) ∪ A3 A1 ∩ (A2 ∩ A3 ) = (A1 ∩ A2 ) ∩ A3 A1 ∪ A2 = A2 ∪ A1 A1 ∩ A2 = A2 ∩ A1 A ∩ (∪j Aj) = ∪j (A ∩ Aj) A ∪ (∩j Aj) = ∩j (A ∪ Aj) } } } (Associative laws) (Commutative ... P (A1 ∩ A2 ) = P (A1 ) − P (A1 )P (A2 ) = P (A1 ) [1 − P (A2 )] = P (A )P(Ac ) = P (A )P (A ) 1 2 Similarly if A = Ac and A = A2 For A = Ac and A = Ac , P (A ∩...

Ngày tải lên: 23/07/2014, 16:21