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1 1 1.1...................................... 1 1.2................................... 5 1.3................................... 7 1.4............................. 9 1.5.................................... 16 1.6...................................... 2 1.7....................................... 24 1.8.................................... 36 1.9.................................... 45 1.1............................. 47 2 53 2.1 (Inverse Transform method)......................... 53 2.1.1........................ 53 2.1.2........................ 59 2.1.3................................ 63 2.2 (Composition method)............................. 66 2.3 (Convolution method).............................. 82 2.4 (Acceptance-Rejection method)...................... 87 2.4.1....................... 87 2.4.2....................... 92 2.4.3................................ 93 2.4.4 (Squeeze method)............................ 13 3 17 3.1 (Normal distribution)...................... 17 3.1.1 Box-Muller Polar............................ 11 3.1.2 Monty Python................................. 114 3.1.3 Odd-Even................................... 12

iv 3.1.4 Ziggurat.................................... 129 3.1.5.......................... 139 3.1.6....................... 143 3.2 (Half Normal distribution).................. 144 3.3 (Log-Normal distribution).................. 153 3.4 (Cauchy distribution).................... 156 3.5 (Lévy distribution)....................... 175 3.6 (Exponential distribution).................... 181 3.7 (Laplace distribution).................... 19 3.8 (Rayleigh distribution)................... 2 3.9 (Weibull distribution).................... 24 3.1 (Gumbel distribution)................... 222 3.11 (Gamma distribution).................... 228 3.12 (Beta distribution)...................... 244 3.13 (Dirichlet distribution)................. 25 3.14 (Power Function distribution)............... 257 3.15 (Exponential Power distribution)............. 26 3.16 (Erlang distribution).................... 281 3.17 χ 2 (Chi-Square distribution)..................... 283 3.18 χ (Chi distribution)......................... 286 3.19 F (F distribution).......................... 289 3.2 t (t distribution)........................... 293 3.21 (Inverse Gaussian distribution).............. 316 3.22 (Triangular distribution).................... 321 3.23 (Pareto distribution).................... 328 3.24 (Logistic distribution)................ 343 3.25 (Hyperbolic Secant distribution)............ 366 3.26 (Raised Cosine distribution).................. 377 3.27 (Arcsine distribution)..................... 39 3.28 (von Mises distribution)............. 394 3.29 (Non-Central Gamma distribution).......... 44 3.3 (Non-Central Beta distribution)............ 48 3.31 χ 2 (Non-Central Chi-Square distribution).......... 416 3.32 χ (Non-Central Chi distribution)............... 419

v 3.33 F (Non-Central F distribution)................ 422 3.34 t (Non-Central t distribution)................. 426 3.35 (Planck distribution).................... 431 4 433 4.1 (Binomial distribution)...................... 433 4.1.1..................................... 435 4.1.2............................ 437 4.1.3 (Condensed table-lookup method)............... 439 4.1.4 (Table plus Square histogram method)..... 444 4.1.5....................... 448 4.1.6 (Table plus Square histogram plus Inverse transform method)............................... 449 4.2 (Geometric distribution)..................... 45 4.3 (Poisson distribution).................... 459 4.4 (Hypergeometric distribution)................ 471 4.5 (Multinomial distribution).................... 484 4.6 (Negative Binomial distribution).............. 486 4.7 (Negative Hypergeometric distribution)......... 494 4.8 (Logarithmic Series distribution).............. 499 4.9 (Yule-Simon distribution).............. 55 4.1 (Zipf-Mandelbrot distribution).... 512 4.11 (Zeta distribution)...................... 514 A 517 A.1....................................... 517 A.1.1............................... 52 A.1.2........................... 53 A.1.3................. 534 A.2....................................... 539 A.2.1............................... 539 A.2.2............................ 543 A.3................................... 554 B 557 B.1................ 557

vi B.2................................ 559 B.3........ 56 C 1/2 563 D 571 D.1................................... 571 D.2................................... 573 D.3................ 575 D.4.............................. 579 D.5................................ 581 E 583 587 591

2 [,1] [,1) (,1) ( ) 2.1 (Inverse Transform method) ( ) ( ) 2-1 1.7 2-2 F(x) 1 U = F(X) F 1 2-3 [,1] X F U [,1] [,1] ( ) 2.1.1 f F F(x) = f(x)dx (2.1) F 2-4 < F(x) < 1 lim F(x) = x 2-1 ( ) ( ) 3 1 2-2 (monotonic increasing function, increasing function) x 1 < x 2 F(x 1 ) F(x 2 ) (non-decreasing function) (monotonic decreasing function, decreasing function) x 1 < x 2 F(x 1 ) F(x 2 ) (non-increasing function) 2-3 1 (single-valued function) f y = f(x) x y 1 x y 2 (multi-valued function) 2-4 (strictly increasing function) x 1 < x 2 F(x 1 ) < F(x 2 ) x 1 < x 2 F(x 1 ) > F(x 2 ) (strictly decreasing function)

54 2 lim x F(x) = 1 2-5 U = F(X) U F X (,1)( < U < 1) F X U 1 1 2-6 F F 1 X = F 1 (U) (2.2) (2.2) U (,1) (U (,1) ) X F(x) ( 2.1) 2.1 F(x) X F(x) 1 U2 x1 U1 x2 x 2.1: 2.1: F(x) (x 1 < x 2 F(x 1 ) < F(x 2 )) < F(x) < 1 ( lim F(x) =, lim F(x) = 1) x x Step1. (,1) U Step2. X = F 1 (U) 2.1 X F ( 2.1) x P(X x) = F(x) 2-7 F P(X x) = P ( F 1 (U) x ) = P (U F(x)) = F(x). 2 F U (, 1) ( ) < F(x) < 1 2-5 1.7 (x 1 < x 2 F(x 1 ) F(x 2 )) F(x) 1 1 1 ( 2-6) (x 1 < x 2 F(x 1 ) < F(x 2 )) < F(x) < 1 ( lim F(x) =, lim F(x) = 1) x x F(x) 1 (x F(x) = x x F(x) = 1 x 1 1 ) 2.1.3 2-6 f 1 1 (1 1 one-to-one function) x 1 x 2 f(x 1 ) f(x 2 ) x 1 x 2 f(x 1 ) f(x 2 ) 2-7 x X X x 1 1 X x F (x) X F (x) X x F (x) X F

2.1. (Inverse Transform method) 55 F F 1 (, 1) U X = F 1 (U) X (,1) U f X Step1. f F F(x) = Step2. F F 1 f(t)dt Step3. (,1) U X = F 1 (U) Step1.,Step2. ( ) ( ) Step3. (,1) Step1.,Step2. ( ) ( ) (,1) [ 2.1] (Exponential distribution) (Exp(θ) ) f Exp(θ) (x) 181 (3.129) f Exp(θ) (x) x ( 1 F Exp(θ) (x) = f Exp(θ) (t)dt = θ exp t ) ( dt = 1 exp x ), (x ) θ θ u = F Exp(θ) (x) x F Exp(θ) 1 (u) = θ ln (1 u), ( u < 1) (2.3) (,1) U 2-8 X = θ ln U (2.4) Exp(θ) X 2-9 (2.4) Exp(4) 2-1 2.2 2-11 2-8 (,1) [, ) [,1) (2.3) 2.1.3 (,1) U 2-9 1 U (,1) 1 U (,1) U 1 U U ( 68 2.6 ) 2-1 ( ) ( ) (frequency table) (histogram) (proportional histogram) ((, ) ) 1 1 ( 1 ) 2-11 [,2) 1 ( ) 1 B

56 2.25.2.15 f(x).1.5 5 1 15 2 x 2.2: Exp(4) [ 2.2] (Exponential-exponential distribution) (EExp(µ, ) ) f EExp(µ,) (x) = 1 ( ) { ( )} x + µ x + µ exp exp exp, ( < x <, < µ <, > ) (2.5) µ (location parameter) (scale parameter) (2.5) F EExp(µ,) (x) = f EExp(µ,) (t)dt = 1 exp { exp ( < x <, < µ <, > ) ( x + µ )}, (2.6) (2.6) F 1 EExp(µ,) (u) = µ + ln { ln (1 u)}, ( < u < 1, < µ <, > ) (2.7) (, 1) U 1 U (, 1) U X = µ + ln ( ln U) (2.8) EExp(µ, ) X µ = = 1 EExp(, 1) (2.5) EExp(µ, ) EExp(, 1) Y X = µ + Y (2.9) EExp(, 1) f EExp(,1) (y) = exp {y exp (y)}, ( < y < ) (2.1) Y = ϕ(x) = X + µ (2.11)

2.1. (Inverse Transform method) 57 ϕ (x) = 1/ g(x) = f EExp(,1) (ϕ(x)) ϕ (x) { ( )} 1 x + µ x + µ = exp exp ( 21 (1.33)) (2.5) EExp(, 1) Y (2.9) EExp(µ, ) X EExp(, 1) Y Exp (1) Z Y = ln Z (2.12) (2.12) (184 (3.153)) f Exp(1) (z) = exp (z) ψ (y) = exp (y) Z = ψ (Y ) = exp (Y ) (2.13) g (y) = f Exp(1) (ψ(y)) ψ (y) = exp {y exp (y)} ( 21 (1.33)) (2.1) Exp (1) Z (2.12) Y (2.12) Y Z 2.2 2.2: EExp(µ, ) X Step1. Exp (1) Z z > Step2. Y = ln (Z) Step3. X = µ + Y * 2.1 3.6 * EExp (, 1) Step3. Y 2.2 Step1. (2.8) (,1) U Exp (1) Z z > (2.4) (,1) U Z = ln U (2.9) (2.12) 2.2 X = µ + Y = µ + ln Z = µ + ln ( ln U)

58 2 (2.8) EExp( 5, 2) (2.8) 2-12 2.3 f(x).2.18.16.14.12.1.8.6.4.2-1 -5 5 1 x 2.3: EExp( 5, 2) f(x) x f(x) x f(x) x 2.2 EExp( 5, 2) 2.4 2.5 2.4 x = 5 x = f(x) F(x) (F(x) = f(t)dt) f(x) x F(x) f(x) x F(x) f(x) x F(x) f(x) x F(x) f(x) x F(x) 2.4 2.5 f(x) x = 5 F(x) f(x) x = F(x) (,1) F(x) F(x) U F 1 F(x) x ( 2.1 ) 2.5 x = 5 F(x) x = F(x) x = 5 x = x x U f(x) X 2-12 [-13,1) 1 ( ) 1 B

2.1. (Inverse Transform method) 59 f(x).2.18.16.14.12.1.8.6.4.2-1 -5 5 1 x 2.4: EExp( 5, 2) F(x) 1.9.8.7.6.5.4.3.2.1-1 -5 5 1 x 2.5: EExp( 5, 2) 2.1.2 X (53 2-1) F(x) = P(X x) = x i x p(x i ) (2.14) (2.14) p X x 1, x 2,..., (x 1 < x 2 < ) (2.14) x 1 < x 2 F(x 1 ) < F(x 2 ) F(X) 1 [,1] U 2-13 2.3 2-13 F(X) 1 1 [,1)

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