Class Notes in Statistics and Econometrics Part 2 pptx

... + t 2 x 2 2! + t 3 x 3 3! + ···(3.10 .20 ) m x (t) = E[e tx ] = 1 + t E[x] + t 2 2! E[x 2 ] + t 3 3! E[x 3 ] + ···(3.10 .21 ) d dt m x (t) = E[x] + t E[x 2 ] + t 2 2! E[x 3 ] + ···(3.10 .22 ) d 2 dt 2 m x (t) ... of binary digits these same questions would be: is the number between 1 and 16?, then: is it between 1 and 8 or 17 and 24 ?, then, is it between 1 and 4 or 9...

Ngày tải lên: 04/07/2014, 15:20

... (18 .2. 24), and Problem 23 7 with y = x to the denom- inator, to get (18 .2. 22) . To prove equation (18 .2. 23) for ˆα, let us work backwards and plug (18 .2. 24) into the righthand side of (18 .2. 23): (18 .2. 25) ... ¯y)  2  (x t − ¯x) 2  (y t − ¯y) 2 . Hint: you are allowed to use (18 .2. 22) . Answer. (18.3 .20 ) R 2 = ˆ β 2  (x t − ¯x) 2  (y t − ¯y) 2 =   (x t...

Ngày tải lên: 04/07/2014, 15:20

... formula for s 2 (i): ( 32. 4.10) s 2 (i) = 1 n − k −1  (n − k)s 2 − ˆε 2 i 1 − h ii  Answer. This merely involves re-writing SSE and SSE(i) in terms of s 2 and s 2 (i). ( 32. 4.11) s 2 (i) = SSE(i) n ... different linear combinations, and to ask: Which linear combination of the eleme nts of ˆ β is affected most if one drops the ith observation? Interestingly and, in h...

Ngày tải lên: 04/07/2014, 15:20

... |T| INTERCEP 1 66 .29 56 029 2 0.770 625 98 86. 028 0.0001 GYD 72 1 0.08686483 0.015 027 57 5.780 0.0001 GYD72L1 1 0.0 125 8635 0. 023 01437 0.547 0.5853 GYD72L2 1 0.0046 125 89 0. 023 21459 0.199 0.8 428 GYD72L3 ... -0.005511693 0. 023 66979 -0 .23 3 0.81 62 GYD72L4 1 -0.0 027 898 62 0. 023 721 00 -0.118 0.9065 GYD72L5 1 0.00 828 0160 0. 023 54535 0.3 52 0. 725 6 GYD72L6 1 -0.001408690...

Ngày tải lên: 04/07/2014, 15:20

... form (66.5.5)  y 1 y 2 y 3 y 4      γ 11 γ 12 γ 13 γ 14 γ 21 γ 22 γ 23 γ 24 γ 31 γ 32 γ 33 γ 34 γ 41 γ 42 γ 43 γ 44     = =  x 1 x 2 x 3 x 4      β 11 β 12 β 13 β 14 β 21 β 22 β 23 β 24 β 31 β 32 β 33 β 34 β 41 β 42 β 43 β 44     +  ε ε ε 1 ε ε ε 2 ε ε ε 3 ε ε ε 4  Answer. Γ ... in the first columns of Γ and B. Let us write these first...

Ngày tải lên: 04/07/2014, 15:20

... xxiii Chapter 2. Probability Fields 1 2. 1. The Concept of Probability 1 2. 2. Events as Sets 12 2.3. The Axioms of Probability 20 2. 4. Objective and Subjective Interpretation of Probability 26 2. 5. Counting ... 737 29 .1. Building the Constraint into the Model 738 29 .2. Conversion of an Arbitrary Constraint into a Zero Constraint 740 29 .3. Lagrange Approach to Constrained Le...

Ngày tải lên: 04/07/2014, 15:20

... e µ x αt+σ 2 x α 2 t 2 /2 e µ y βt+σ 2 y β 2 t 2 /2 = e (µ x α+µ y β)t+(σ 2 x α 2 +σ 2 y β 2 )t 2 /2 , which is the moment generating function of a N(αµ x + βµ y , α 2 σ 2 x + β 2 σ 2 y ).  We will ... is 1 b−a  • b. 2 points Compute E[x] Answer. E[x] =  b a x b−a dx = 1 b−a b 2 −a 2 2 = a+b 2 since b 2 − a 2 = (b + a)(b − a).  • c. 2 points...

Ngày tải lên: 04/07/2014, 15:20

... VARIANCE 22 1 Answer. Go back to throwing a coin twice independently and define A = {HH, HT }; B = {T H, HH}, and C = {HH, T T }, and x 1 = I A , x 2 = I B , and y = I C . They are pairwise indepen- dent, ... E[y] 2 (E[x 2 ] − E[x] 2 ) + (E[x 2 ] − E[x] 2 )(E[y 2 ] − E[y] 2 ) = E[x 2 ] E[y 2 ] − E[x] 2 E[y] 2 = E[(xy) 2 ] − E[xy] 2 .  8.6. Conditional Expe...

Ngày tải lên: 04/07/2014, 15:20

... < 2 . For the latter we need 1 2 σ 2  ∞ 0 re −r 2 /2 2 dr = 1 2 , set q = r 2 /2 2 , then dq = 1 σ 2 r dr, and the integral becomes 1 2  ∞ 0 e −q dq.  • d. 1 point Are r and φ independent? Answer. ... N(µ, σ 2 ) is (two notations, the second is perhaps more modern:) (10.1.4) f x (x) = 1 √ 2 σ 2 e − (x−µ) 2 2σ 2 = (2 σ 2 ) −1 /2 exp  −(x − µ) 2 /2 2...

Ngày tải lên: 04/07/2014, 15:20

... = y 1 + y 2 2 ( 12. 3.10) y 1 − ¯y = y 1 − y 2 2 ( 12. 3.11) y 2 − ¯y = − y 1 − y 2 2 ( 12. 3. 12) (y 1 − ¯y) 2 + (y 2 − ¯y) 2 = (y 1 − y 2 ) 2 4 + (y 1 − y 2 ) 2 4 ( 12. 3.13) = (y 1 − y 2 ) 2 2 = σ 2  y 1 − ... 1 point Show further that s 2 u = 1 2  3 i=1 (y i − ¯y) 2 = 1 2 (z 2 2 + z 2 3 ). (There is a simple trick!) Conclude from this that s 2...

Ngày tải lên: 04/07/2014, 15:20