A Course in Mathematical Statistics phần 3 docx

... 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 ... exists an open interval containing x and contained in G. Without loss of generality, such intervals may be taken to be centered at x. It follows from this definition that an open inter...

Ngày tải lên: 23/07/2014, 16:21

... Moment Generating Function and Factorial Moment Generating Function of a Random Variable The ch.f. of an r.v. or an r. vector is a function defined on the entire real line and taking values in the ... the normal curve, the horizontal axis, and the abscissas 1 and 3. Clearly, the correction, given by the area bounded by the normal curve, the horizontal axis and the abscissas 1.5 and 3....

Ngày tải lên: 23/07/2014, 16:21

... 5928 .3 5981.1 6022.5 6055.8 60 83. 3 6106 .3 0.990 0.995 237 15 239 25 24091 24224 2 433 4 24426 0.995 0.500 1 .30 45 1 .32 13 1 .33 44 1 .34 50 1 .35 37 1 .36 10 0.500 0.750 3. 335 2 3. 3526 3. 3661 3. 3770 3. 3859 3. 3 934 ... 24. 935 29.615 32 .671 35 .479 38 . 932 41.401 22 26. 039 30 .8 13 33. 924 36 .781 40.289 42.796 23 27.141 32 .007 35 .172 38 .076 41. 638 44.181...

Ngày tải lên: 23/07/2014, 16:21

... 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 example, take I = 3, J = 5 and let Y 11 = ... 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 s...

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 ... previous values of m, n and 1 − α , we have F 13, 12;/0.025 = 3. 238 8 and F 12, 13; 0.025 = 3. 1 532 , so that the corresponding interval approximately is equal to 0 31 71 3 238 8 12 2...

Ngày tải lên: 23/07/2014, 16:21

... 4 34 8 13 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 past ... 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...

Ngày tải lên: 23/07/2014, 16:21

... 01Ω 30 2 12 Point Estimation and its variance attains the Cramér–Rao bound. 12.4.2 Refer to Exercise 12 .3. 5 and investigate whether the Cramér–Rao bound is attained. 12.4 .3 Refer to Exercise 12 .3. 6 ... () ∈ [] = () ∈ [] 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...

Ngày tải lên: 23/07/2014, 16:21

... , 9 .3 Linear Transformations of Random Vectors 239 9.1 The Univariate Case 235 9 .3 Linear Transformations of Random Vectors 235 9.2.11 Let X r be an r.v. distributed as t r . Then show that: EX ... defined as follows: Syxyxyx Syxyxyx Syxyxyx Syxyxyx Syxyxyx Syxyxy 1 23 1 1 2 2 3 3 132 1 1 2 3 3 2 2 131 221 33 231 12 23 31 31 2 13 21 3 2 32 1 1 3 2 2 :, , :, , :,, :,, :,...

Ngày tải lên: 23/07/2014, 16:21

... 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 Refer ... 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 a cert...

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...

Ngày tải lên: 23/07/2014, 16:21