A Course in Mathematical Statistics phần 2 pps

... the class C by: CAA=× ∈ ∈ {} ×= () ∈∈ {} AAA A AA ss sAs A 121 12 2 12 121 12 2 ;, , ,; , .where 76 3 On Random Variables and Their Distributions As will be seen in Chapter 4, the p.d.f.’s f 1 and ... remaining suit; iii) At least two of the cards are aces; iv) Two cards are aces, one is a king, one is a queen and one is a jack; v) All five cards are of the same suit. 2. 4...

Ngày tải lên: 23/07/2014, 16:21

... Then fy rr rrr y y r y r Y r r rr () () / (/) // = + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − −− Γ ΓΓ 1 2 1 2 1 2 11 12 122 2 21 2 2 2 2 1 1 21 r r r y r ye 2 1 1 1 1 2 2 1 21 21 2 →∞ − − − ⎯→⎯⎯ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Γ (/) (/) / since ΓΓ rr r r r r r 12 2 2 221 22 12 1 2 1 + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎯→⎯⎯ [] →∞ −/(/) by ... 32. 73...

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... μ μρσσ μσ 12 2 2 1 2 11 22 11 2 2 2 1 122 12 22 2 1 2 2 +− () () − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− () −− () − () +− () x x x xxx x. Hence fxx x xx x XX 12 12 12 2 2 11 1 2 12 1 12 2 22 2 2 1 21 1 21 2 , , exp , () = − − − () ⎧ ⎨ ⎪ ⎩ ⎪ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ −− () − () + − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ ⎥ ⎫ ⎬ ⎪ ⎭ ⎪ πσ ... soils and fertilizers. Comparison of a certain brand of gasoline...

Ngày tải lên: 23/07/2014, 16:21

... 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 ... interval (b 1 , b 2 ], say, then γ = 0 again and any C in (b 1 , b 2 ] can be chosen without affecting the level of the test. This is so because P Y b b Gb Gb ab ab ab ab θθ 0 12...

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... y ydy yy ydy Y y 1 1 1 22 1 1 2 1 2 2 2 2 2 22 1 2 2 2 0 22 () −∞ ∞∞ =− + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− + () ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ∫∫ ππ exp exp . Set y yt y t y 1 2 2 2 2 2 1 2 1 2 2 1 + () == + , so that and 2 2 11 0 22 1 2 22 1 2 ydy dt y ydy dt y t= + = + ∈∞ [ ) ,,,.or Thus ... 0, and − ρ if cd < 0; ii) (cX + dY, cX − dY)′ ϳ N(c μ 1 + d μ 2 , c μ 1 − d μ 2 , τ 2 1...

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... 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 experiment according to a (random) ... sequential procedure: As long as a < λ n < b, take another observation, and as soon as λ n ≤ a, stop sampling and accept H and as soon as λ n ≥ b, stop sampling and reject...

Ngày tải lên: 23/07/2014, 16:21

... 01Ω 3 02 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 ... θ () ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 . (6) 12. 4 The Case Where Complete Sufficient Statistics Are Not Available or May Not Exist 29 5 28 4 12 Point Estimation 28 4 Chapter 12 Point Estimatio...

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... 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 () =+ () + () [] =− () ++ () = () − () [] + () + () [] = ()() − ()() [] + () 22 22 12 21 11 12 21 22 1 2 () − ()() [] =+ () + () = ()() EY ... Then d dt Mt d dt eteEX X t t t t t t t () ==+ () == () = + = + = 0 2 0 2 2 0 22 22 μ σ μ σ μσ μ , and d dt Mt d dt te ete EX X X t t t...

Ngày tải lên: 23/07/2014, 16:21

... Fx 12 2 1 12 222 2 22 1 ,, where F is the joint d.f. of X 1 , X 2 and F 2 is the d.f. of X 2 . By letting h 1 , h 2 → 0 and assuming that (x 1 , x 2 )′ and x 2 are continuity points of f and ... (Hint: Observe that the exponent in the joint p.d.f. of X 1 and X 2 may be written as follows: 1 21 2 2 21 2 11 1 2 11 1 22 2 22 2 2 11 2 1 2 2 2 2 2 2 2...

Ngày tải lên: 23/07/2014, 16:21

... AAA 121 21 2 ∪=+ −∩ () . Hence PA A PA PA A A PA PA PA A 12 1 21 2 121 2 ∪ () = () +−∩ () = () + () −∩ () , since A 1 ∩ A 2 ⊆ A 2 implies PA A A PA PA A 21 2 2 1 2 −∩ () = () −∩ () . (C7) P is subadditive; ... n n nn n nn 23 2 10 21 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +⋅⋅⋅+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =−− relationships characterizing the independence of...

Ngày tải lên: 23/07/2014, 16:21