Introduction to Probability phần 2 doc

... Ratio 1 1 . 922 1.084 2 2 1.919 1.0 42 3 6 5.836 1. 028 4 24 23 .506 1. 021 5 120 118.019 1.016 6 720 710.078 1.013 7 5040 4980.396 1.011 8 40 320 399 02. 395 1.010 9 3 628 80 359536.873 1.009 10 3 628 800 3598696.619 ... of radius 1 /2, which occurs with probability p = π(1 /2) 2 π(1) 2 = 1 4 . 2. L > √ 3 if |r| < 1 /2, which occurs with probability 1 /2 − (−1 /2) 1 −...

Ngày tải lên: 09/08/2014, 23:20

... occurs with probability p = π(1 /2) 2 π(1) 2 = 1 4 . 2. L> √ 3if|r| < 1 /2, which occurs with probability 1 /2 −(−1 /2) 1 −(−1) = 1 2 . 3. L> √ 3if2π/3 <α<4π/3, which occurs with probability 4π/3 ... z E Figure 2. 19: Calculation of F Z . 20 40 60 80 100 120 0.005 0.01 0.015 0. 02 0. 025 0.03 f (t) = (1/30) e - (1/30) t Figure 2. 20: Exponential density wi...

Ngày tải lên: 04/07/2014, 10:20

... = 7 /2 and σ 2 = V (X) = 35/ 12. Thus, E(S 420 ) = 420 · 7 /2 = 1470, σ 2 (S 420 ) = 420 · 35/ 12 = 122 5, and σ(S 420 ) = 35. Therefore, P (1400 ≤ S 420 ≤ 1550) ≈ P  1399.5 − 1470 35 ≤ S ∗ 420 ≤ 1550.5 ... .0000 1.0 .3413 2. 0 .47 72 3.0 .4987 .1 .0398 1.1 .3643 2. 1 .4 821 3.1 .4990 .2 .0793 1 .2 .3849 2. 2 .4861 3 .2 .4993 .3 .1179 1.3 .40 32 2.3 .4893 3.3 .4...

Ngày tải lên: 09/08/2014, 23:20

... is P =      1 2 3 0 4 1 0 1 /2 0 1 /2 0 2 1 /2 0 1 /2 0 0 3 0 1 /2 0 0 1 /2 0 0 0 0 1 0 4 0 0 0 0 1      . From this we see that the matrix Q is Q =   0 1 /2 0 1 /2 0 1 /2 0 1 /2 0   , and I ... −1 /2 0 −1 /2 1 −1 /2 0 −1 /2 1   . Computing (I − Q) −1 , we find N = (I − Q) −1 =   1 2 3 1 3 /2 1 1 /2 2 1 2 1 3 1 /2 1 3 /2   . From the middle row of...

Ngày tải lên: 09/08/2014, 23:20

... lead. b 2k,2n = 1 2 k  j=1 f 2j α 2k−2j,2m−2j + 1 2 m−k  j=1 f 2j α 2k,2m−2j = 1 2 k  j=1 f 2j u 2k−2j u 2m−2k + 1 2 m−k  j=1 f 2j u 2k u 2m−2j−2k = 1 2 u 2m−2k k  j=1 f 2j u 2k−2j + 1 2 u 2k m−k  j=1 f 2j u 2m−2j−2k = 1 2 u 2m−2k u 2k + 1 2 u 2k u 2m−2k , where ... return to the origin at time 2k is given by f 2k 2 2k u 2n−2k 2 2n−2k = f 2k u 2n−2k 2 2n . If we...

Ngày tải lên: 09/08/2014, 23:20

... Smith Unconditional probability b bb b g g g b 1/4 1/8 1/8 1/8 1/8 1/4 1/4 1/4 1/4 1/4 1 1 /2 1 /2 1 /2 1 /2 1 bg gb gg b bb b b 1/4 1/8 1/8 1/4 1/4 1/4 1 1 /2 1 /2 bg gb Conditional probability 1 /2 1/4 1/4 Figure 4.13: Tree for Example 4 .26 . 148 CHAPTER 4. CONDITIONAL PROBABILITY .001 can not .01 .95 .05 + - .001 0 .05 .949 + - .051 .949 + - .981 1 0 can not .001 .05 0 .949...

Ngày tải lên: 04/07/2014, 10:20

... 3138 12 3043 13 26 90 14 24 23 15 25 56 16 24 56 17 24 79 18 22 76 19 23 04 20 1971 21 25 43 22 26 78 23 27 29 24 24 14 25 26 16 26 24 26 27 23 81 28 20 59 29 20 39 30 22 98 31 20 81 32 1508 33 1887 34 1463 35 ... pgs. 115- 120 . 5 .2. IMPORTANT DENSITIES 20 5 Integer Times Integer Times Integer Times Chosen Chosen Chosen 1 26 46 2 2934 3 33 52 4 3000 5 33...

Ngày tải lên: 04/07/2014, 10:20

... + Y ) 2 ) −(a + b) 2 = E(X 2 )+2E(XY )+E(Y 2 ) −a 2 − 2ab − b 2 . Since X and Y are independent, E(XY )=E(X)E(Y )=ab. Thus, V (X + Y )=E(X 2 ) −a 2 + E(Y 2 ) −b 2 = V (X)+V (Y ) . ✷ 24 6 CHAPTER ... any random variable with E(X)=µ, then V (X)=E(X 2 ) −µ 2 . Proof. We have V (X)=E((X − µ) 2 )=E(X 2 − 2 X + µ 2 ) = E(X 2 ) 2 E(X)+µ 2 = E(X 2 ) −µ 2 . ✷ U...

Ngày tải lên: 04/07/2014, 10:20

... 1 0 0. 02 0.04 0.06 0.08 0.1 0. 12 0.14 0 0 .2 0.4 0.6 0.8 1 0 0. 02 0.04 0.06 0.08 0.1 0. 12 0 0 .2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0 .2 0 .25 0 0 .2 0.4 0.6 0.8 1 0 0. 025 0.05 0.075 0.1 0. 125 0.15 0.175 n=10 n =20 n=40 n=30 n=60 n=100 Figure ... E(X i )=  1 0 xdx= 1 2 , σ 2 = V (X i )=  1 0 x 2 dx −µ 2 = 1 3 − 1 4 = 1 12 . Hence, E  S n n  = 1 2 , V  S n n  = 1...

Ngày tải lên: 04/07/2014, 10:20

... return to the origin at time 2k is given by f 2k 2 2k u 2n−2k 2 2n−2k = f 2k u 2n−2k 2 2n . If we sum over k, we obtain the equation u 2n 2 2n = f 0 u 2n 2 2n + f 2 u 2n 2 2 2n + ···+ f 2n u 0 2 2n . Dividing ... paths of length (2m −2j) which have exactly 2k line segments above the t-axis is b 2k,2m−2j . Therefore, we have b 2k,2m = 1 2 k  j=1 f 2j b 2k−2j,2m−2j + 1 2 m...

Ngày tải lên: 04/07/2014, 10:20