A Course in Mathematical Statistics phần 1 ppt

... n n nn n nn 23 2 10 21 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +⋅⋅⋅+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =−− relationships characterizing the independence of A j , j = 1, , n and they are all necessary. For example, for n = 3 we will have: PA A A PA PA PA PA A PA PA PA A PA PA PA A PA PA 12 3 1 2 3 12 1 ... following examples: Let S = {1, 2, 3, 4}, P( {1} ) = ···= P({4}) = 1 4 ,...

Ngày tải lên: 23/07/2014, 16:21

... 515 Table 1 (continued) p nk1 /16 2 /16 3 /16 4 /16 5 /16 6 /16 7 /16 8 /16 16 14 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 0.9997 15 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 16 1. 0000 ... 9.488 11 .14 3 13 .277 14 .860 5 6.626 9.236 11 .0 71 12.833 15 .086 16 .750 6 7.8 41 10.645 12 .592 14 .449 16 . 812 18 .548 7 9.037...

Ngày tải lên: 23/07/2014, 16:21

... ee JJIIJ JJIIJ IJ 11 1 21 2 1 11 1 21 2 1 11 , , ; , , ; ; , , , , ; , , ; ; , , ; , ; , , ′ ′ ′ββ μα α β β , and X′ = 11 00 0 10 0 0 11 00 0 010 0 11 00 0 000 01 1 010 0 10 0 0 10 10 0 010 0 ⋅⋅⋅ ⋅⋅ ⋅ ⋅⋅⋅ ... is, there are no interactions present); and estimation of σ 2 . By setting Y e = () = () = YY YY YY ee ee ee K J JK IJ IJK K J JK IJ IJK JI 11 1 11 1 1 1 1 11 1...

Ngày tải lên: 23/07/2014, 16:21

... . 1 1 1 1 1 1 1 1 1 0 10 01 1 1 0 10 01 θ θ θθ θθ θ θ θθ θθ (40) Next, Z fX fX X 1 11 01 1 10 01 1 0 1 1 1 1 = () () = − () − () + − − log log log , θθ θθ θ θ so that EZ i ii1 10 01 1 0 1 1 1 1 01= − () − () + − − = θ θθ θθ θ θ log ... replacing α , β , a and b by α ′, β ′, a and b′, respectively, in (29) and also taking into consideration (30),...

Ngày tải lên: 23/07/2014, 16:21

... (See also Figs. 13 .14 and 13 .15 ; F n 1, m 1 stands for an r.v. distributed as F n 1, m 1 .) The power of the test is easily determined by the fact that 1 1 1 1 11 1 2 2 2 1 1 2 2 1 2 1 2 1 σ σ τ YY ... test.) 13 .6.3 Five resistance measurements are taken on two test pieces and the observed values (in ohms) are as follows: xxxxx yyyy y 12 345 12 345 0 11 8 0 12 5 0...

Ngày tải lên: 23/07/2014, 16:21

... justi- fiable 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 several desirable ... p r j j r x r x j j r x r x r x r r r θθ 1 1 1 1 11 1 1 1 11 1 , , ! ! ! ! , () = ⋅⋅⋅ = ⋅⋅⋅ − −⋅⋅⋅− () = = −− ∏ ∏ − where n =∑ r j =1 x j . Then log , , log ! ! log log log...

Ngày tải lên: 23/07/2014, 16:21

... as Negative Exponential with parameter λ = 1. Then: Exercises 233 SJ SJ SJ SJ SJ SJ 12 3 12 3 2 31 2 31 132 13 2 312 312 213 213 3 21 3 21 100 010 0 01 1 0 01 100 010 1 100 0 01 010 1 010 0 01 100 1 010 10 0 0 01 1 0 01 010 10 0 1 :: :;: ::. == ... consider the following result. Let Z 1 , , Z k be independent N(0, 1) , and set Y k Z k Z k Z YZZ YZZZ Y kk Z k...

Ngày tải lên: 23/07/2014, 16:21

... EY EY 1 2 11 12 21 22 11 21 12 22 11 22 12 21 11 21 12 22 11 22 12 21 11 21 12 22 11 () =+ () + () [] =− () ++ () = () − () [] + () + () [] = ()() − ()() [] + () 2222 12 21 11 12 21 22 1 2 () − ()() [] =+ () + () = ()() EY ... probability space (S, A, P) and let A 1 , A 2 be events. Set X 1 = I A 1 , X 2 = I A 2 and show that X 1 , X 2 are independ...

Ngày tải lên: 23/07/2014, 16:21

... f(x 2 |x 1 ) ≥ 0 and fx x fx fx x fx fx xx 21 11 12 11 11 11 1 22 () = () () = () ⋅ () = ∑∑ ,, fx x dx fx fx x dx fx fx 21 2 11 12 2 11 11 11 1 () = () () = () ⋅ () = −∞ ∞ −∞ ∞ ∫∫ ,. In a similar fashion, ... Y XY EXY 1 1 2 2 12 12 12 12 12 Cov , . From the Cauchy–Schwarz inequality, we have that ρ 2 ≤ 1; that is 1 ≤ ρ ≤ 1, and ρ = 1 if and only if YX=+ −...

Ngày tải lên: 23/07/2014, 16:21

... ∩ () +⋅⋅⋅ + PAPABAPABABA PA B A B A cc cc cc cc n c n c n 1 11 2 11 223 11 1 = () +∩ () () +∩ () ∩ () () +⋅⋅⋅+ ∩ () ⋅⋅⋅ ∩ () () +⋅⋅⋅ () + PA PA B PA PA B PA B PA PA B PA B PA cc cc cc cc n c n c n 1 11 ... 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 d...

Ngày tải lên: 23/07/2014, 16:21