2 R A B,, : A B,,.,,,.,,., (random variable),, X Y Z..,., ( 1.1),. 1.1 A B A B A, B ; A B A = B A B A B A B (intersection) A B A B (union) A B A B = A

Transcription

1 ,?,,,,,.,, (random phenomenon),., (random experiment), trial(,, experiment trials).,, (sample space), Ω ;, (sample point), ω,, ω 1, ω 2,, ω n., (random event, ), (subset).., Ω, (sure event);, (empty set), (impossible event).,, ;,.,,.,.,, 9 11,.,,, ;, ;,.

2 2 R A B,, : A B,,.,,,.,,., (random variable),, X Y Z..,., ( 1.1),. 1.1 A B A B A, B ; A B A = B A B A B A B (intersection) A B A B (union) A B A B = A B (disjoint) A B (, mutually exclusive) A c Ā A (complement), A + Ac = Ω A A c A B (difference) A B Ω A 1, A 2,, A n, A 1 A 2 n A n = Ω, A i =Ω, A 1, A 2,, A n Ω (partition), A 1, A 2,, A n i=1 (exhaustive). A 1 A 2 A n n A i., A B,,., : n x i = x 1 + x x n i=1 n x i = x 1 x 2 x n i=1,, (counting technique). (addition principle) (multiplication principle). :, n, 1 m 1, 2 m 2,, n m n, i=1

3 1 3 m 1 + m m n. : n, m,, m n. :,..,,. 1.1 ( ) (factorial),, : n! = n (n 1) (n 2) (1.1), 0! = 1. (double factorial), n (n 2) 4 2, n!! = n (n 2) 3 1, n ; n (1.2) 0!! = 1 R, factorial()., 10! ( ) factorial(10) (permutation) n (without replacement) r(r n) ( ). n P r, P r n A r n(a Arrangement ). :, n P n = n!. np r = n! (n r)! (1.3) n P r,,. P r n A r n.,,,. 1.3 ( ) (combination) n ( ) r(r n) n ( ), n C r C r n., r n r(r n),. :, n C 0 = n C n = 1. nc r = n P r r! = n! (n r)!r! (1.4),.,.. R choose(n, k), n C k., 10 C 5

4 4 R choose(10, 5) 252. R,,., 10 P 5, 10 P 5 = 10 C 5 5!, : choose(10, 5) * factorial(5) (probability)?,. (Leroy Folks) : ( ).,,,,,.,,,,..,.,.,. :.,., ( )., (classical probability), : 10 5,? 10/15 = 2/3. ( ),, ( )..,. 1.4 ( ) : (1) ; (2) ; (3) A [ P (A)] P (A) = A A = (1.5) (1.6) : [M].,. :, 1987: 55.

5 1 5.,, Blaise Pascal ( ) Pierre- Simon Laplace ( ). A (favorable outcomes),, (outcomes of interest).,,,., n, n. : ( ), :,.,?,.,., ;, ( ) ). 6, 6 ( ) ( 6. 1, 365 ; 2, 1, = 364.,, P 6 / R choose(365,6) * factorial(6) / 365^6 R ( ),. 1.2 (k < r)? n r, k,,,,,, ( A), P (A c ) = 1 P (A).,,, ( ) n (k) (n r).. k ( ) n r P (A c ) = 1 P (A) = 1 k ( ) n k 1.3 ( ) k(k a + b). a b,,

6 6 R. X = { k }. (1).,, a + b. a + b a + b, (a + b)!. k k, b ; a + b 1 a + b 1 ( ), (a + b 1)!.,, X b(a + b 1)!. P (X) = b(a + b 1)! (a + b)! = b a + b (2)..,,,., a + b. ( ) a + b, b a + b ( b b,, ), b,. k k, b 1 ( ) a + b 1 a + b 1 b 1,, b 1.. ( ) a + b 1 (a + b 1)! b 1 (b 1)!a! P (X) = ( ) = = b a + b (a + b)! a + b b b!a!. a b,, k. k ( ) k,.,,, ( ),.,.,. 1.4 ( ) (Monty Hall problem), Let s Make a Deal, (Monty Hall). :,,,.,,,.. :?., (, 50% )., 1/3;

7 1 7 ( 2/3),,., 2/3.,,, 2/3., Monty Hall Problem,.,,.,.,, :, ,.,, ;,,.,,,,.,,. (relative frequency).,,,. (empirical probability),. 1.5 ( ), : (1) A ( ) ; (2) n, m A, f n (A) = m n ; (3), n, f n (A) p. p,. (3) : lim n f n(a) = p. f n (A) n p, p, p, p., (Head, H) (Tail, T)., 100 H:T=50:50, :51 ;,.,, : n, f n (A) p., : lim P ( f n(a) p ɛ) = 0 n 2006 Keith Chen(2008) Journal of Personality and Social Psychology How Choice Affects and Reflects Preferences: Revisiting the Free-Choice Paradigm,. Amos Tversky( ) Daniel Kahneman( ),. : (2013, ), (2012, ).

8 8 R ɛ ; P ( ). : p?, 0,.,,.,,.,,., 80%,. 80%,, 80 % ,.,,,, (,,, )., :., : A. : 80%.,. (subjective probability),.,, (prior probability), (posterior probability).,,.,, ( ).,,.,.,,, 20,.,, 20,,,., :, 1?, (, 2012) (, 2013), (, 2015), (Allen B. Downey) : Python (, 2015), (John K. Kruschke) : R BUGS (, 2015),.

9 1 9 1/ = 0.,.,,.,.., 1.,?,,,,, ( ) 2 1, π = π/4. 2, (geometric probability). 1.6 ( ) : (1) Ω, ( ) S Ω ; (2) Ω ; (3) A Ω, S A, A P (A) = S A S Ω (1.7),,,,.,. 0, 1., 0? 1?,.,.,, 1 0, ;,, 1 1,.,.,,,, :00 10:00, 10 min, 10 min. 9:00 10:00,. 9:00 ( : min), (x), (y),, 1.. A = { }. x y 1, 0 x, y 1. 6,, ( 1.1). P (A) = = 11 36

10 10 R d(d > 0), c(c < d),. Buffon (Buffon s Needle Problem). y, θ ( 1.2). 1.2 Buffon : 0 y d, 0 θ π 2, S Ω = dπ 2 ( 1.3). ( A) y c sin θ. : 2 S A = π 0 c 2 sin θ dθ = c 2 cos θ π 0 = c

11 1 11 P (A) = S A S Ω = c dπ 2 = 2c dπ 1.3 Buffon ( c = 2 cm) ( d = 1 cm) π., n, m. P (A) m/n. m n 2c dπ = π 2nc dm. (Monte Carlo Simulation),,,. 1.7 R Buffon, animation., Buffon. R,. library(animation) oopt = ani.options(nmax = ifelse(interactive(), 500, 2), interval = 0.05) par(mar = c(3, 2.5, 0.5, 0.2), pch = 20, mgp = c(1.5, 0.5, 0)) buffon.needle() buffon.needle(redraw = FALSE) ani.options(oopt),?buffon.needle, R.

12 12 R ?,? Andrey Nikolaevich Kolmogorov ( ) 1933., ( ) ( ) E, Ω. E A ( ), P (A), : (1) : A, P (A) 0; (2) : Ω P (Ω) = 1; (3) (Countable Aadditivity): A 1, A 2, i j, A i A j = (i, j = 1, 2, ) P (A 1 A 2 ) = P (A 1 ) + P (A 2 ) + P (A) A. (probability Axioms)., ( A 1, A 2,, A n, n ).,.,,,.,.,,,.,,.,..,,.,.,. (, ),.,,,., :.,.,,,,,..,,,.,,., ;. [M].,,. :, 1987: 62.

13 1 13, ;., : (1) P (Ω) = 1 (2) P ( ) = 0 (3) P (A c ) = 1 P (A) (4) (addition rule): P (A B) = P (A) + P (B) P (A B) 1.3, (conditional probability),., A P (A), B, A,, P (A B), the probability of A given B, given., P (A) (unconditional probability). (die).,. A 4, P (A) = 1/2, B, B., , B, A P (A B) = 1/3. P (B) = 1/2, P (A B) = 1/ ( ) A, B E, P (B) > 0, P (A B) B A, P (A B) = P (A B) P (B) (1.8) (1.8) : P (A B) = n(a B) n(b) (1.9) : n(a B) A B ; n(b) B. n(a B) n(b) = n(a B) n(ω) n(b) n(ω) = P (A B) P (B) : n(ω) Ω.

14 14 R (1.9), B Ω B;, B, A, ( ).,,, ;,,., ( ). (1.8) : A, B, P (A B) = P (B)P (A B) (1.10) P (A B) = P (A)P (B A) (1.11) ( ): ( n ) P A k = P (A 1 )P (A 2 A 1 )P (A 3 A 1 A 2 ) P (A k A 1 A 2 A k 1 ) (1.12) k= , 2, 2, 1, 2 ( ). A= 1 B= 2.,, A, 2, 8.,, 4, 4, P (B A) = 4 C 2 8C 2 = 3 14, P (B A). R 1.9 ). choose(4,2)/choose(8,2) n 2n, n (.. Ω., 1, 2,, 2n (, 2n ), 2k 2k 1 k = 1, 2,, n,. 2n!, 1 1 2n, 2 2n 1 ;, 2 1 2n 2, 2 2n 3 ;., [2n(2n 1)] [(2n 2)(2n 3)] (2 1) = 2n!

15 A n., 1, 2,, n. A k k(k = 1, 2,, n)., A A k n, A = A k. k=1 P (A 1 ). 1,, 1. 2n 1. P (A) = 2n 1. P (A 2 A 1 ). 1, n 1, 2n 2., 1, 2(n 1) 1 = 1 2n 3, P (A 2 A 1 ) = 2n 3., : P (A k A 1 A 2 A k 1 ) = 1, k = 3, 4,, n 2n (2k 1) P (A) = P (A 1 )P (A 2 A 1 )P (A 3 A 1 A 2 ) P (A k A 1 A 2 A k 1 ) 1 = 2n 1 n = k= n (2n 1)!!, 0.5, 0.2,?, : = 0.1. ( :?)? :., (independence). 1.9 ( ) : A B, P (A B) = P (A) P (B A) = P (B) (1.13), A B (dependent).. : A, B ;.., A, B ;.,.

16 16 R 1.10 ( ) A, B, C P (A B) = P (A)P (B) P (A C) = P (A)P (C) P (B C) = P (B)P (C) A, B, C (1.14) P (A B C) = P (A)P (B)P (C) (1.15) A, B, C,., (1.10) : P (A B) = P (A)P (B) (1.16), A B., (1.16). (1.13) (1.16) : A={ 5 } B={ 6 } C={ 7 } : (1) A B? (2) A C? (1) A={(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}, 6. P (A) = 6 36 = 1 6. B={(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}, 5. A B 6, 1 5., {(5, 1)}. P (A B) = 1 5., P (A B) P (A), A B. (2) C={(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}, 6. A C 7, 1 5., {(5, 2)}. P (A C) = 1 6., P (A C) = P (A), A C.. :,,? :, (1.13) (1.16) (,

17 1 17 ).,,. 60 ( 100),, 59 60, a b,., k b a + b. : (1) 1 2? (2), 1,, b a + b? (1). A 1 1, A 2 2.,, P (A 1 ) = P (A 2 ) = b a + b. P (A 2 A 1 ). A 2 A 1 1, 2. a + b 1, b 1 ( 1 ). P (A 2 A 1 ) = b 1 a + b 1., P (A 2 A 1 ) P (A 2 ) (2). 1., (1), 2 P (A 2 A 1 ) = b 1 a + b 1 b a + b., 1, 2 P (A 2 A c b 1) = a + b 1 b. 1, a + b..,?,,.,,. :,,,,,?.,,,, ;,,,., ( ) b a + b.., A B, A B, A B,.. ; ;.?,

18 18 R.,,.,,.,,,,,.,,.,,..,,, A, B.,,,., ( ).,,, (Bayes Formula). 1.1 ( ) : 0 < P (B) < 1, A P (A) = P (B)P (A B) + P (B c )P (A B c ) (1.17) n : B 1, B 2,, B n, B i = Ω, P (B i ) > 0, i = 1, 2,, n, A P (A) = i=1 n P (B i )P (A B i ) (1.18) i=1 (1.17), (1.18) %, 10%.. S=, L=,, P (S) = 3/10, P (L) = 1/10, P (L S) = 10P (L S c ). P (L S). P (L) = P (S)P (L S) + P (S c )P (L S c ) = P (S)P (L S) P (Sc )P (L S), 1 10 = 3 10 P (L S) + 1 ( ) P (L S) 10 P (L S), P (L S) = 10/37..

19 ( ) : P (A) > 0, P (B) > 0, P (B A) = P (B)P (A B) P (A) (1.19) (1.19) P (A) P (A) = P (B)P (A B)+P (B c )P (A B c ). n : B 1, B 2,, B n, B i = Ω, P (A) > 0, P (B i ) > 0, i = 1, 2,, n, i=1 P (B i A) = P (B i)p (A B i ) P (A) = P (B i)p (A B i ) n P (B i )P (A B i ) i=1 (1.20) (1.19)., P (A B) = P (A)P (B A) = P (B)(A B) P (A)P (B A) = P (B)(A B) = P (B A) = (1.20) P (B)P (A B) P (A), 40, 1%., 80%;, 9.6% ( ).,,? A=, B=,, P (B) = 0.01, P (A B) = 0.8, P (A B c ) = P (B A). 1.4., (1.19), P (B)P (A B) P (B A) = P (A) P (B)P (A B) = P (B)P (A B) + P (B c )P (A B c ) = (1 0.01)

20 20 R 1.4..,,, 95%, 75%., 1% (prior probability). Tversky Kahneman, Sabrina. 3/10,, 1/5, 1/10, 2/5., 1/4;, 1/3 1/12;.. Sabrina. :,? A, B 1, B 2, B 3, B 4. B 1, B 2, B 3, B 4. : P (B 1 ) = 3 10 P (B 2 ) = 1 5 P (A B 1 ) = 1 P (A B 2 ) = P (B 1 A). [ (1.20)], P (B 1 A) = P (B 1)P (A B 1 ) 4 P (B i )P (A B i ) = i=1 P (B 3 ) = 1 P (B 4 ) = P (A B 3 ) = 1 P (A B 4 ) = = 1 2

21 , n, A B. 3. n 0 n 1, , 6, : (1) ; (2) 4 ; (3) : n, n 1, n 2,, n k k, n 1 + n n k = n. k, i n i, i = 1, 2,, k.. 5.,. : :,,.,,.,, :,,.,, ( ).,,,.,.,,, :.,,,?. 6. [0, 1], A B,. A 1, B 2,. 8. a > 0, x, y, 0 < x < a, 0 < y < a, xy < a , , ,., , :,. p, 1 p.,.

22 22 R 12. 5%, 0.25%, 22:21., (1)? (2),? 14..,,., , 0.3, 0.1., 0.7. (1). (2),. 16.,,. A,. A. : A, 2/3., 1/2, 1., A,. :? 17.,? , 2 2, 3 3, 4 1, 2, 3, , : P (A i )={, i} (i = 1, 2, 3) : (1) A 1, A 2, A 3?(2) A 1, A 2, A 3?(3),? 18. : ( n P k=1 A k ) = P (A 1 )P (A 2 A 1 )P (A 3 A 1 A 2 ) P (A k A 1 A 2 A k 1 ) 19. :. 20. R 20 30, ( ). 365,.

23 , Paul Erdos( ).,.,, ,,.,.,.,.,, 1, 0.,,.,. 2.1 ( ) E, Ω, ω Ω, X(ω), Ω X(ω), X = X(ω) (random variable)., X, X f(x).., Ω, ( )., ( ), ( )., ; ( ),,.,, ;,.., , 1, 0. A

24 24 R, A ,, X 1, 2,, X, X min.., Y, Y [0, 5] A = 0, 2.2 X = 5 5, 2.3 Y > 3 3min., (,, ), 2.3, ( ), (discrete random variable). (a, b), a, b, (continuous random variable).,,.,,.,,.,,.., X x, :. 2.3 ( ) X, x, F (x) = P (X x), < x < X (distribution function of random variable),, X F (x), (is distributed as). X F X (x), X,., X F (x) X x, (cumulative distribution function, CDF). (,,, ),,. F (x), X x, x. X F (x), X.,.,., x. 2.3,

25 2 25 [0, 5]., (, ). X < 0, P (X < 0) = 0; x 5, X < x, P (X x) = 1; 0 x < 5, P (X x) = x/5., F (x) : 0, x < 0; x F (x) =, 0 x < 5; 5 1, x 5, (, ),.,.,. (1) : F (x) (, ), x 1 < x 2, F (x 1 ) F (x 2 ); (2) : x, 0 F (x) 1, F () = lim F (x) = 0 x F ( ) = lim x F (x) = 1 (3) : F (x) x, a lim F (x) = F (a), F (a + 0) = F (a) x a +,. F (x). ( ),,., F (x). F (x), X., a < b, : P (a < X b) = F (b) F (a) P (a X b) = F (b) F (a 0) P (X b) = 1 F (b 0) P (X > b) = 1 F (b) P (a < X < b) = F (b 0) F (a) P (a X < b) = F (b 0) F (a 0),. F (x) a, b, F (a 0) = F (a), F (b 0) = F (b)

26 26 R, F (a 0) = lim x a F (x) P (a < X < b) = P (a < X b) = P (a X < b) = P (a X b) cm.,. X,. x < 0, X x, F (x) = 0., x 10, X x, F (x) = 1. 0 x 10., P (0 X x) = P (X x) = F (x) = kπx 2, k. x = 10, P (X 10) = F (10) = 1., k = 1/100π., X : F (x) = 1 100π πx2 = x2 0, x < 0; 1 F (x) = 100 x2, 0 x < 10; 1, x 10, F (x), a X < b, a < X b ( ) X, x i (i = 1, 2,, n, ), X P (X = x i ) = p(x i ) = p i, i = 1, 2,, n, (2.1) X (probability mass function, PMF),,. ( 2.1),. 2.1 X x 1 x 2 x n P (X) p 1 p 2 p n

27 2 27,.. :, p i 0;,. p i = 1.. : F (x) = P (X x) = p i, i = 1, 2,, n, (2.2) x i x i=1, ( ). X x i, X x i p i., F (x), F (x) x i P (X = x i ) = p i.,.,.,., (0,1) x 1, x 2,, P (X = x i ) = 1, i = 1, 2, 2i,.,,. [ : (0,1), ],,,, : (1)X ; (2)X. 1,2,3,4, , X 3 X. (1),. X 3, 4, 5. X. 5 3, 5 C 3 = 10. 3,, {1, 2, 3}; 4, 3 : {1, 2, 4}, {1, 3, 4}, {2, 3, 4}; 5, 6, {1, 2, 5},{1, 3, 5},{1, 4, 5}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}. P (X = 3) = 1 10 X 3 6 = 0.1, P (X = 4) = = 0.3, P (X = 5) = = 0.6 X P (X)

28 28 R (2), 0, x < 3; 0.1, 3 x < 4; F (x) = 0.4, 4 x < 5; 1, x 5, c, P (X = c) = 1., (degenerate distribution). 1, x c; F (x) = 0, x < c 2.1.4, X (a, b), x i P (X = x i ) 0., 1,., X, X, X.,,,. 2.5 ( ) f(x), x F (x) X, F (x) = x f(t) dt (2.3) X, F (x), f(x) X (probability density function, PDF),. f(x) : (1), f(x) 0; (2), f(x) dx = 1. f(x),. [ f(x) x ], f(t) F (x).. x,, {x < X x + h} P (x < X x + h) = F (x + h) F (x)., [F (x + h) F (x)] /h x h (x, x + h),. h 0, F (x + h) F (x) lim = F (x) = f(x) h 0 h x., 1,., 1. [f(x)h = f(x) dx],.

29 2 29,,.,,..,,, F (x) = f(x)., P (a < X < b),. X (a, b], x = a, x = b, y = f(x) x., F (x) = f(t) dt = 1., P (a < X < b) = P (a X < b) = P (a < X b) = P (a X b). 0.,, (0, a), X. (0, a),,. X. X F (X) f(x)., F (x) x f(x). x < 0, F (x) = P (X x) = 0; x a, F (x) = P (X x) = 1. 0 x < a, F (x) = P (X x) = P (0 X x) = kx, k. x = a, F (x) = 1, k = 1/a., F (x) = x/a., X 0, x < 0; x F (x) =, 0 x < a; a 1, x a F (x),. x < 0 x > a, f(x) = 0; 0 < x < a, f(x) = 1/a. x = 0 x = a, F (x),., ( ),.,.,,,.,., 0,., X 1, 0 x < a; f(x) = a 0, (0, a) (uniform distribution),, X U(0, a).

30 30 R 2.7 0, x < 0; x F (x) =, 0 x < 1; 2 1, x 1 : (1)F (x)?(2) F (x), X?. (1) :.. F (x), F (x). (2) X, f(x) F (x) = x f(t) dt, f(x) dx = 1. f(x),.,,, F (x) = f(x). F (x) x, 1 F (x) = f(x) = 2, 0 < x < 1; 0, x < 0 x > 1 f(x), x = 1, f(1),, f(x) dx = 1., f(x) dx = 1/2 1., F (x) X,.,.,,., ,,.,. (, ). (expectation, expected value), (mean).,, ( ) ( ) X P (X = x i ), i = 1, 2,, n,, x i p i, x i p i X i=1 i=1

31 2 31,, E(X), µ. E(X) = x i p i (2.4) i (i = 1, 2, 3,, n), E(X) = i=1 n x i p i (2.5) i=1,.,,,,. x i,,.,,.,. 2.7 ( ) xf(x) dx, E(X) = X f(x),.,.. xf(x) dx (2.6) xf(x) dx. x i p i,, f(x), f(x) x i=1, X x f(x) x, (, ), (improper integral, ).. :,,,..,..,,., 3 800, 3 800,,, , ;, , 70.

32 32 R 2.2.2,.,.,,.?,,, A B,.,,,,.., X E(X) = X µ,,., [ E(X µ) = 0, ].,, X µ,., ( 20,,, ), : (X µ) 2., (X µ) 2, X µ ; (X µ) 2 ( 0), X µ., (X µ) 2,. X, µ, (X µ) 2. (X µ) 2,.,, ( ) X 2 E(X 2 ), [X E(X)] 2 X (variance), Var(X) = E[X E(X)] 2 (2.7) (standard deviation, SD). σ 2 ( sigma ), σ (σ Σ ). D(X), D Deviation. X σx 2 σ X.,. 2.9 ( ) X [x E(X)] 2 p i, X ; Var(x) = i=1 n (2.8) [x E(X)] 2 p i, X i=1 (2.7) (2.8), ( [X E(X)] 2 ).

33 ( ) X : Var(X) = [x E(X)] 2 f(x) dx (2.9) (2.8) (2.9),,.,, ;,,..,, ,,.,,.,,.. : (1) c ( ), E(c) = c, ; (2) E(cX) = ce(x); (3) E(aX + b) = ae(x) + b, a, b ( ); (4) E(X ± Y ) = E(X) ± E(Y ); (5) g(x), h(x), E[g(X) ± h(x)] = E[g(X)] ± E[h(X)]; (6) X Y, E(XY ) = E(X)E(Y );. (1) (2) (3), (4) (5),,.. 2.1,.,. (6),.,,, ( ) X P (X i ) = p i f(x), X Y = g(x) E(Y ) = E[g(X)] = g(x i )p i, X ; i=1 (2.10) E(Y ) = E[g(X)] = g(x)f(x) dx, X

34 34 R.,. 2.8 X : E(X 2 ). X 0 1 P (X) 1 p p 2.1, X 2,. Y = f(x) = X 2, Y X., X, Y ;, X, Y.,., Y, f(x) = X 2, X. X 2 : X P (X) 1 p p E(X 2 ) = 1 p + 0 (1 p) = p. 2.1, : E(X 2 ) = 0 2 (1 p) p = p. 2.9 X : E(X 2 ). X P (X) X 0 1 P (X) , X 2 : P (X 2 = 1) = 0.5, P (X = 1) + P (X = 1) = ,,. E(X 2 ) = = , : E(X 2 ) = ( 1) = 0.5.,. 2.1, g(x) g(x)..,, n, n. X,.

35 2 35 X i (i = 1, 2,, n) 1, i ; X i = 0, i n X = X i. i=1 E(X i ) = 1 P (X i = 1) + 0 P (X i = 0) = 1 n n E(X) = E(X i ) = n 1 n = 1. i=1,, (1) Var(c) = 0, 0; (2) Var(X + c) = Var(X); (3) Var(aX + b) = a 2 Var(X); (4) Var(X) = E(X 2 ) [E(X)] 2 ; (5) X Y, Var(X ± Y ) = Var(X) + Var(Y ). (1). (2) (3). (3) (4). (3)., Var(aX + b) = E[(aX + b) E(aX + b)] 2 = E[aX + b ae(x) b] 2 = a 2 E[X E(X)] 2 = a 2 Var(X) (4)., Var(X) = E[X E(X)] 2 = E{X 2 2X E(X) + [E(X)] 2 } = E(X 2 ) 2E[X E(X)] + E[E(X)] 2 = E(X 2 ) 2E(X) E(X) + [E(X)] 2 = E(X 2 ) [E(X)] 2 (4).

36 36 R (5), X Y, X + Y X Y,,, (3) Var( Y ) = Var(Y ). X Y.,, X( : ),. : (1) ; (2) Y = 60(13 X),,. X P (X) (1), E(X)=11.4( ), Var(X)= 0.84=0.917( ).,,. (2) E(Y ) = E[60(13 X)] = 60E(13 X) = E(X) = 96( ). Var(Y ) = Var[60(13 X)] = 60 2 Var(13 X) = 3 600Var(X) = = 55( ) X e x 2, x < 0; F (x) = 1/2, 0 x < 1; e 1 2 (x 1), x 1., E(X) = e x 2, x < 0; f(x) = 0, 0 x < 1; 1 4 e 1 2 (x 1), x 1 xf(x) dx = = = 1 Var(X), E(X 2 ). 0 x ex 2 dx + 1 [ ] 1 x 4 e 1 2 (x 1) dx E(X 2 ) = x 2 f(x) dx = = = x 2 e2 2 dx + 1 x 2 [ 1 4 e 1 2 (x 1) ] dx Var(X) = E(X 2 ) [E(X)] 2 = = 6.5.

37 , 10..,. X,. X = X i (i = 1, 2,, 10) 1, i ; X i = 0, i n X i. i 1/10, i i=1 9/10. i,,, (9/10) 20, 1 (9/10) 20. X i X i 0 1 ( ) 9 20 ( ) 9 20 P (X i ) ( ) [ 20 ( ) ] 20 ( ) E(X i ) = = E(X) = [ n E(X i ) = 10 1 i=1. ( ) ] 20 9 = ,,.,, : p, n, k?.. n k,, ( ) n, (k n k ) p k (1 p) n k. k ( ) n p k (1 p) n k.. k, (binomial distribution) : (1) n ;

38 38 R (2) ; (3), ; (4). (success),, (failure).,, 4, ( ) X ( ) n P (X = k) = p k (1 p) n k, k = 0, 1, 2,, n (2.11) k X, X B(n, p). : n ; p ; k n. n, p (parameter). n p,. X 0, 1, 2,, n n ,, p = 0.5, ; p < 0.5, ( ); p > 0.5, ( ) E(X) = np Var(X) = np(1 p) (2.12)

39 2 39.,, (Bernoulli distribution). n = 1, : X 0 1 P (X) 1 p p, X = 0, X = 1., p, p(1 p). (Bernoulli trial)., n, n., X i (i = 1, 2,, n), 1, i ; X i = 0, i i p. X = X 1 + X X n. X B(n, p)., : E(X) = E(X 1 + X X n ) = E(X 1 ) + E(X 2 ) + + E(X n ) = np Var(X) = Var(X 1 + X X n ) = Var(X 1 ) + Var(X 2 ) + + Var(X n ) = np(1 p),,., : 20%. 100, 120,. X= 120, X B(120, 0.8). P (X > 100) = 1 P (X 100) = R 1-pbinom(100, 120, 0.8), 100, 120, R, R, ( ) X (2.13) P (X = k) = e λ λk, k = 0, 1, 2, (2.13) k! λ > 0, X λ (poisson distribution), X P (λ). X π(λ) X Poisson(λ).

40 40 R (2.13), k=0 λ λk e k! = e λ k=0, λ k k! = e λ e λ = 1 (2.14) e x = 1 + x + x2 2! + + xk k! + = k=0 x k k! (2.15) x = λ. (2.15). 1 : λ. µ, : (µ) (σ 2 ) λ.. X λ,,, E(X) = = = λ = λ k=0 k=1 λ λk ke k! λ λk ke k! k=1 m=0 e λ λk 1 (k 1)! λ λm e m! ( k = 0 0, 1 ) [ m = k 1, (2.14) ] = λ (2.16) E(X 2 ) = = k=0 k 2 λ λk e k! kλe λ λk 1 (k 1)! k=1 m=k 1 ===== λ = λ k=0 m=0 λ λm me = λe(x) + λ λ λm (m + 1)e m! m! + λ k=0 λ λm e m! = λ 2 + λ (2.17) Var(X) = E(X 2 ) [E(X)] 2 = λ 2 + λ λ 2 = λ (2.18)

41 2 41 λ (rate) (intensity), ( )., Simeon-Denis Poisson ( ),, n,.,,,,,,. λ. λ, ; λ, 0, ;,. λ = 20, B(n, p), n p,, (,,, ).., X P (λ 1 ), Y P (λ 2 ), Z = X + Y P (λ 1 + λ 2 ). X Y..,

42 42 R,. X B(n, p), n, p 0 np λ,.., ( n p ), λ = np < 5,.,,.,, (Stochastic Process).,.,, (count data), ( ), (Poisson Regression) : p(0 < p < 1).,.. X=, P (X = k), k = 1, 2,.. X = k k, k 1. P (X = k) = (1 p) k 1 p. (geometric distribution). P (X = k) = (1 p) k 1 p ( ) X P (X = k) = (1 p) k 1 p, k = 1, 2,, n, (2.19) X, X G(p)., p ; k. ( p). ( 2.3).. : (1) ; (2), ; (3) ; (4)., n,. E(X) = 1 p Var(X) = 1 p p 2 (2.20)

43 q = 1 p,, E(X) = k(1 p) k 1 p = p kq k 1 = p i=1 = p d dq,. ( ) q k n=0 i=1 = p d dq i=1 ( q 1 q ) = :. : 2.3 ( ) i=1 dq k dq p (1 q) 2 = 1 p x n = 1, x < 1. 1 x X G(p), m n, P (X > m + n X > m) = P (X > n) (2.21)., ( A) (X). X > m m A. n, A, X > m + n., m A, n A, n A, m, m. 2.3.

44 44 R P (X > n) = (1 p) k 1 (1 p)n p = p = (1 p)n 1 (1 p) k=n+1 P (X > m + n X > m) = = P ({X > m + m} {X > m}) P (X > m) P (X > m + n) P (X > m) = = (1 p) n = P (X > n). (1 p)m+n (1 p) m, X r, X (r, p) (negtive binomial distribution), X Nb(r, p), P (X = k) = ( ) k 1 (1 p) k r p r (2.22) r 1 :, ( ( ) p). k 1 k 1 r 1, p r 1 (1 p) k r, r 1 p, (2.22). r = 1,., E(X) = r p Var(X) = r 1 p p 2 (2.23) 2.15, 11:30 12:30, 20%... (1),,,? (2) 10,? (3) : 5%,,.,. 100, 10,, : 20%?,. (1) X., X G(0.2). E(X) = 5, 5.

45 2 45 (2) 10, P (X > 10) = 1 P (X 10) = , 100, 11, 10., R. R 1-pgeom(9, 0.2),, pgeom(k-1, p) k 1. r = 1., R, pgeom()(r ) pgeom(q, p), q.,,. (3) 0.2, X 100, Y B(100, 0.2). P (Y 10) = 10 k=0 ( ) k k = < 0.05 k R pbinom(10,100,0.2)., 0.2, ,, :! ( ;, ) , , (hyper-geometric distribution) : N M, n, k? N,,., : P (X = k) = ( )( ) M N M k n k ( ) N, k = 0, 1, 2,, r (2.24) n : r {M, N}, M N, n N, n, N, M., (2.24)., r P (X = k) = 1., (2.24) X, X H(n, N, M). k=0

46 46 R E(X) = n M N Var(X) = nm(n M)(N n) N 2 (N 1) (2.25),,.,..,.,,, N n, M/N.,, : ( )( ) M N M ( ) k n k n ( ) p k (1 p) n k, p = M N k N n (2.26),, 1/10.,. (relatively large population) ( ) X 1 f(x) = b a, a < x < b; 0, (2.27) X (a, b) (uniform distribution), X U(a, b)., [a, b]. ( ) E(X) = a + b 2 Var(X) = (a b)2 12 X U(a, b), a < c < c + l < b c c + l, P (c < X < c + l) = c+l c 1 b a dx = l b a X (a, b),.. : X (a, b),?

47 2 47,,, X U(0, 20). X 10, 10,, P (X > 10) = 0.5(?). Y 10 X 10. X B(10, 0.5). ( ) 10 1 P (Y = 0) = = R 1-pbinom(0, 10, 0.5) ( ) X λe λx, x 0; f(x) = 0, x < 0. (2.28) X (exponential distribution), X Exp(λ), λ > 0. (skewed distribution, ), ( 2.4).,. 2.4 : E(X) = 1 λ Var(X) = 1 λ 2 (2.29)

48 48 R E(X) = xf(x) dx = = xe λx x d( λe λx ) e λx dx = 1 λ e λx = 1 0 λ,. θ = 1/λ, θ. 1 e λx, x 0; F (x) = 0, x < 0 (2.30). 2.4 X Exp(λ), s t, P (X > s + t X > s) = P (X > t) (2.31).,. X ( : ), s P (X > s), t P (X > s + t X > s), t P (X > t), s P (X > t) = 1 P (X t) = 1 (1 e λt ) = e λt P (X > s + t X > s) = = P ({X > s + t} {X > s}) P (X > s) P (X > s + t) P (X > s) = e λt = P (X > t) = e λ(s+t) e λs t [0, t] N(t) λt, T. N(t) P (λt), λt (λt)k P [N(t) = k] = e, k = 0, 1, 2, k! T, {T > t} [0, t], {T > t}={n(t) = 0}, P (T > t) = P [N(t) = 0].

49 2 49 t < 0, F T (t) = P (T t) = 0. t 0, F T (t) = P (T t) = 1 P (T > λt (λt)0 t) = 1 P [N(t) = 0] = 1 e = 1 e λt. 0! λ, T Exp(λ) min), ( : 1 f(x) = 5 e x 5, x 0; 0, x < 0 (2.32), 10min,. 10, Y. P (Y 2). p = P (X > 10) = 1 P (X 10) = 1 (1 e 2 ) = e 2. Y B(10, e 2 ). P (Y 2) = 1 P (Y 1) = R 1-pbinom(1,10,exp(-2)) (normal distribution), (Gauss distribution), (Friedrich Gauss, ).,.,,., ; ;, ; ( ) X f(x) = 1 2πσ e (x µ)2 2σ 2, < x < (2.33) X, X N(µ, σ), µ, σ > 0. F (x) = 1 2πσ x e (t µ) 2 2σ 2 dt, < x < (2.34) F (x). X N(µ, σ 2 ),,. X N(µ, σ), ;, σ 2, σ. ( ),.

50 50 R : µ σ.,. (normal curve) (error curve),., ( 2.5) : (1) (bell-shaped) (symmetric) (uni-modal, ), x = µ; (2) x, (asymptote) x ; (3), (median) (mode) (mean) ; (4) µ ± σ (inflection points). 2.5, (µ) (σ).,,, ( 2.6). 2.6 ( ),, ( ),, ( 2.7). ( ),,,,

51 2 51,. 2.7,.,? ( ) X f(x) = 1 2π e x2 2, < x < (2.35) X (standard normal distribution), X N(0, 1), 0, 1. Z, Z N(0, 1). F (x) = 1 2π x e t2 2 dt, < x < (2.36). 2.5 ( ) X N(µ, σ), Z = X µ, Z N(µ, σ). σ (standardization). X Z F X F Z, f X (x) f Z (z). ( X µ F Z (z) = P (Z z) = P σ ) z = P (X µ + zσ) = F X (µ + zσ). F Z (z) = F X (µ+ zσ) z, f Z (z) = d dz F X(µ + zσ) = f X (µ + zσ) σ = 1 2πσ e (µ+zσ µ)2 2σ 2 σ = 1 2π e z2 2, Z N(0, 1). Z z (z-score), z (quantile) (standardised score) X N(µ, σ), E(X) Var(X).

52 52 R,. Z = X µ, 2.5, σ Z N(0, 1). E(Z) = z 1 2π e z2 2 (integrand), 0( ). E(Z) = 0. Var(Z) = E(X 2 ) [E(X)] 2 = E(X 2 ) = dz z 2 1 2π e z2 2 = 1 z 2 e z2 1 2 dz = z d ( e z2 2 ) 2π 2π = 1 [ z( e z2 2 ] ) e z2 2 dz 2π = 1 e z2 1 2 dz = 2π = 1 2π 2π dz Γ( Gamma) : 0. t/ 2 x, e x2 dx = π/2,,, 0 ( ) e ( t 2 ) 2 t d = e t2 2 dt = π 2 e t22 dt = 2 0 e t2 2 dt = 2 2 π 2 = 2π,. Γ,. Gamma Beta,.,. 0, 1. : E(X) = E(µ + Zσ) = E(µ) + 0 = µ; Var(X) = Var(µ + Zσ) = σ 2 Var(Z) = σ 2,,,.,,,.

53 (1) P ( 2 Z 2), Z N(0, 1); (2) P (Z 2), Z N(0, 1); (3) P ( 2 X 2), X N(4, 2); (4) P (X 20), X N(15, 10). (1) R: pnorm(2)-pnorm(-2)=0.9545; R : pnorm(q, mean =, sd = ), mean=0, sd=1, ; q (quantile), P (X < x) x., pnorm(2,0,1) pnorm(2) P (X 2), X N(0, 1). R,. (3). (2) R: 1-pnorm(2)= (3) P ( 2 X 2) = P (X 2) P (X < 2) ( X 4 = P 2 4 ) ( X 4 P = P (Z 1) P (Z < 3) < 2 4 ) 2.,,.,. R: pnorm(2,4,2)-pnorm(-2,4,2)= (4) R: 1-pnorm(20,15,10)= X( : kg) N(µ, σ). P (X 70) = 0.5, P (X 60) = 0.25, : (1) µ σ; (2) 10, 5 65kg. ( X µ (1) P (X 70)=P < 70 µ ) ( =P Z 70 µ ) = 0.5, 70 µ = σ σ σ σ 0, µ = 70. ( ) P (X 60) = P Z = z , σ = σ = σ R z : qnorm(0.25).,r z, qnorm(p, mean =, sd = ),,. (2) Y 10, 65kg. Y B(10, p), p = P (X > 65) = P (Y 5) = 1 P (Y 4) =

54 54 R ) 2.22 X N(172, 6)( cm). (1) 0.01? ( (2) 185 cm, (1) a cm, P (X a) = 1 P (X a) = 0.01, X N(172, 6). P (X a) = 0.99 a. R qnorm(0.99, 172, 6) (2) Y 100, Y B(100, p), p = P (X 185) = P (Y 2) = , R : 2.23 X N(µ, σ). : (1) P (µ σ X µ + σ); (2) P (µ 2σ X µ + 2σ); (3) P (µ 3σ X µ + 3σ). pbinom(2,100,0.0151)., (1), P (µ σ X µ + σ) = P (X µ + σ) P (X µ σ) ( = P Z µ + σ µ ) ( P Z µ σ µ ) σ σ = P (Z 1) P (Z 1) = P (µ 2σ X µ + 2σ) = ; P (µ 3σ X µ + 3σ) = ,, 68% ; 95% ; 99.7%.., N(163, 6)( : cm), : 68% 157 cm 169 cm, 95% 151 cm 175 cm, 99.7% 145 cm 181 cm,.,,,, (empirical rule). 3., t χ 2 F,.

55 , X, X g(x). g(x). X g(x), X. X : X x 1 x 2 x n P (X) p 1 p 2 p n g(x), : X g(x 1 ) g(x 2 ) g(x n) P (X) p 1 p 2 p n [g(x i ) = g(x k ), i, k = 1, 2, ], g(x) X, Y = X Z = X 2 + X. X P (X) Y 0, 1, P (Y = 0) = P (X = 0) = 0.3 P (Y = 1) = P (X = 1) + P (X = 1) = 0.7 Y : Y 0 1 P (Y ) , Z 0, 2, P (Z = 0) = P (X = 1) + P (X = 1) = 0.5 P (Z = 2) = P (X = 1) = 0.5 : Z 0 2 P (Z)

56 56 R , X U(0, 1). Y = X n. 1, 0 < x < 1; f X (x) = 0, 0, x < 0; F X (x) = x, 0 x < 1; 1, x 1 Y (0,1). 0 < y < 1, F Y (y) = P (Y y) = P (X n y) = P (X y 1 n ) = FX (y 1 n ) = y 1 n y, Y 1 f Y (y) = n y 1 n n, 0 < y < 1; 0, (2.37) 2.26 X f X (x), Y = X 2. Y 0. y 0, F Y (y) = P (Y y) = P (X 2 y) = P ( y X y) = F X ( y) F X ( y) y, Y 1 f Y (y) = 2 y [f X( y) + f X ( y)], y 0; 0, y < X f X (x), Y = X. (2.38), y 0, F Y (y) = P (Y y) = P ( X y) = P ( y X y) = F X (y) F X ( y), Y f X (y) + f X ( y), y 0., Y = g(x), Y,., f Y (y) = df Y (y). dy F Y (y) = P (Y y) = P (g(x) y) = g(x) y f X (x) dx (2.39)

57 2 57 Y.,,,., X, f X (x). Y = g(x), y = g(x), x = h(y), Y = g(x), : (a, b) y = g(x). f X [h(y)] h (y), a < y < b; f(x) = 0, y = g(x). Y a, F Y (y) = 0; Y b, F Y (y) = 1. a < Y < b, F Y (y) = P (Y y) = P [g(x) y] = P (X h(y)) = F X [h(y)] = h(y) f X (x) dx (2.40) y = g(x), h = g(y) (?). h (y) > 0. y, f Y (y) = f X [h(y)] h (y) y = g(x),, h (y) [, :, f Y (y), ]. a ,. 2.7 X N(µ, σ), Y = ax + b N(aµ + b, a σ), a, b, X Y (, ). a > 0, y = ax + b(a 0), x = h(y) = y b a, h (y) = 1 a. 2.6 ( ) y b f Y (y) = f X [h(y)] h (y) = f X 1 a a = 1 e ( y b a µ)2 2σ 2 1 2πσ a 1 = e [y (aµ+b)] 2 2(aσ) 2 2π(aσ) N(aµ + b, aσ). a < 0, y = ax + b(a 0), x = h(y) = y b a, h (y) = 1 a f Y (y) = e [y (aµ+b)] 2 2(aσ) 2 2π( a σ)

58 58 R N(aµ + b, a σ)..,, X N(0, σ), Y = X. Y = X (, ), y = x x = y, 2.7 Y N(0, σ). X Y.,,. 2.6,, k 2.18 ( ) X, k, µ k = E(X k ) (2.41) k (original moment), ν k = E[X E(X)] k (2.42) k (central moment). E[(X c) k ], c = 0, c = E(X).,,. (moment).., ( ), ( ),. k, ν k = E[X E(X)] k = E(X µ 1 ) k (2.43) 4 : ν 1 = 0; ν 2 = µ 2 µ 2 1; ν 3 = µ 3 3µ 2 µ 1 + 2µ 3 1; ν 4 = µ 4 4µ 3 µ 1 + 6µ 2 µ 2 1 3µ 4 1 (2.44), X N(0, σ), 4 ( ) µ 1 = ν 1 = 0; µ 2 = ν 2 = σ 2 ;

59 2 59 µ 3 = ν 3 = 0; µ 4 = ν 4 = 3σ 4.,,, ,., ;,. :,.,,., ( ) CV(X) = Var(X) E(X) (2.45) X (coefficient of variation).,.,,, ( )., F (x) α x, x α. x α (quantile)., (quantile regression),, ( ) X f(x), F (x), 0 < α < 1, F (x α ) = xα f(x) dx = α (2.46) x α X α,. α, (percentile). α = 0.5, x 0.5 (median), x 0.5 F (x α ) = x0.5 f(x) dx = 0.5, α = , x 0.25 x 0.75 (first quartile) (third quartile).

60 60 R α z α, N(µ, σ), α x α x α = µ + σ z α., 1 F (x α ) = x α f(x) dx = α (2.47) x α., ,., (skewness) ( ) E[X E(X)]3 β s = [E(X E(X)) 2 ] 3 2 (2.48) X (coefficient of skewness)., (skewedness). 0, 0. β s,. β s > 0, (positively skewed), (right skewed),, ; β s < 0, (negatively skewed), (left skewed),.,,,., ;, ( 2.8) (kurtosis).

61 ( ) β k = X (coefficient of kurtosis). E[X E(X)]4 [E(X E(X)) 2 ] 2 3 (2.49). 0. β k < 0,, (heavy tail); β k > 0,, (light tail). k,.,,., β s β k 0,.. 2.7,.,,,.,,.,,,,, (n ) X 1 (ω), X 2 (ω),, X n (ω) Ω = {ω} n, X(ω) = (X 1 (ω), X 2 (ω),, X n (ω)) (2.50) n ( n ), (random vector)., ( ) n x 1, x 2,, x n, n {X 1 x 1 },{X 2 x 2 },, {X n x n } F (x 1, x 2,, x n ) = P (X 1 x 1, X 2 x 2,, X n x n ) (2.51) n (jointly culmulative probability distribution function),.. (X, Y ), F (x, y) = P (X x, Y y) {X x}, {Y y}.,,

62 62 R a < b, c < d P (a < X b, c < y d) = F (b, d) F (a, d) F (b, c) + F (a, c) 0 (2.52) F (x, y) ( ) (X, Y ) (x i, y j ), p ij = P (X = x i, Y = y j ), i = 1, 2,, j = 1, 2, (2.53) (jointly probability mass function),, (X, Y ) ( ) f(x, y), (X, Y ) F (x, y) F (x, y) = x y f(u, v) dv du (2.54) (X, Y ) (jointly continuous random variable), f(x, y) (X, Y ) (joint probability density function). F (x, y), f(x, y) = 2 F (x, y)(2.55) x y,. G, {(X, Y ) G} G P ((X, Y ) G) = f(x, y) dx dy G, f(x, y) G,,.,,.. X Y F X (x) F Y (y), (X, Y ) (marginal distribution), F X (x) X, F Y (y) Y. F (x, y) y, {y < }, F (x, ) = lim y F (x, y) = P (X x, Y < ) = P (X x) = F X(x) (2.56) F (, y) = P (X <, Y y) = P (Y y) = F Y (y).

63 ( ) (X, Y ) P (X = x i, Y = y j ), j P (X = x i, Y = y j ) = P (X = x i ), i = 1, 2, (2.57) j=1 X., i Y. P (X = x i, Y = y j ) = P (Y = y j ), j = 1, 2, (2.58) i=1 (X, Y ) f(x, y), F X (x) = F (x, ) = F Y (y) = F (, y) = x y ( f X (x) = f Y (y) = ) f(u, v) dv du = ) f(u, v) du dv = ( x y f X (u) du f Y (v) dv f(x, y) dy (2.59) f(x, y) dx (2.60), f X (x) f Y (y) X Y., (X, Y ), (, ),.,, X Y,., ( ) n (X 1, X 2,, X n ) F (x 1, x 2,, x n ), X i F (x i ), x 1, x 2,, x n F (x 1, x 2,, x n ) = n F (x i ) (2.61) i=1 P (X 1 x 1, X 2 x 2,, X n x n ) = n P (X i x i ) (2.62) i=1

64 64 R X 1, X 2,, X n, X 1, X 2,, X n., n x 1, x 2,, x n ( ) n x 1, x 2,, x n, P (X 1 = x 1, X 2 = x 2,, X n = x n ) = n f(x 1, x 2,, x n ) = f Xi (x i ), i=1 n P (X i = x i ), i=1 (2.63), X Y f(x, y) = f X (x)f Y (y) (2.64) : f(x, y) (X, Y ) ; f X (x) f Y (y) X Y ,. (X, Y ), X (conditional ditribution), Y X., X, Y, X Y. 1.7m,, ( ) P (Y = y j ) = p j = (X, Y ) p ij = P (X = x i, Y = y j ), i = 1, 2,, j = 1, 2, p ij > 0 y j, i=1 p i j = P (X = x i Y = y j ) = P (X = x i, Y = y j ) P (Y = y j ) = p ij p j, i = 1, 2, (2.65) Y = y j X. X = x i Y ( ) (X, Y ) f(x, y), f X (x), f Y (y), f X (x) > 0 x, X = x Y F (y x) = f(y x) = y f(x, y) f X (x). f(x, v) dv (2.66) f X (x) (2.67),,.

65 ( ) f Y (y) = f X (x) = f X (x)f(y x) dx; (2.68) f Y (y)f(x y) dy (2.69) 2.32 ( ) f(y x) = f Y (y)f(x y) f Y (y)f(x y) dy (2.70) f(x y) ( ) value), : P (Y = y j X = x), j E(Y X = x) = yf(y x) dy, (conditional expected E(X Y = y). (X, Y ) ; (X, Y ) E(Y X = x) x. x, E(Y X = x),,.. g(x) = E(Y X = x),,,. (X, Y ). f Y X (y x) = f Y (y) f X Y (x y) = f X (x) (2.71) X Y.., X N(µ 1, σ 1 ), Y N(µ 2, σ 2 ), X, Y, X + Y N(µ 1 + µ 2, σ1 2 + σ2 2 )... X, Y,., X N(0, 1), Y = X, Y N(0, 1), X + Y 0., 0., Var(X) + Var(Y ) = σ σ 2 2, Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X, Y ), Cov(X, Y )..

66 66 R, ax N(aµ, a σ), a, : n, a 1 X 1 + a 2 X a n X n N(µ 0, σ 0 ) (2.72), n n µ 0 = a i µ i, σ0 2 = a 2 i σi 2 (2.73) i=1 i=1, X N(3, 1), Y N( 2, 2), X, Y, 2X 3Y 10 N(2, 40) ,.,,., (n ). 2.8 ( ) (X, Y ), Z = g(x, Y ) ( ) P (X = x i, Y = y i ), i j E[g(X, Y )] = g(x, y)f(x, y) dx dy, ; (2.74),. X 1, X 2,, X n, E(X 1 X 2 X n ) = E(X 1 )E(X 2 ) E(X n ); Var(X 1 ± X 2 ± ± X n ) = Var(X 1 ) + Var(X 2 ) + + Var(X n ) X 1, X 2,, X n, E(X 1 ± X 2 ± ± X n ) = E(X 1 ) ± E(X 2 ) ± ± E(X n ).., ( ) X Y (covariance). E[(X E(X))(Y E(Y ))], Cov(X, Y ) = E[(X E(X))(Y E(Y ))] (2.75) ρ XY = Cov(X, Y ) = Cov(X, Y ) (2.76) Var(X) Var(Y ) σ X σ Y X Y (coefficient of linear correlation), ρ.

67 2 67 X X E(X), Y Y E(Y ). : (1) X E(X) Y E(Y ) ( X Y, E(X) E(Y ) ), Cov(X, Y ) > 0, ρ > 0, X Y (positive linear correlation); (2) X E(X) Y E(Y ) ( X Y ), Cov(X, Y ) < 0, ρ < 0, X Y (negative linear correlation); (3) Cov(X, Y ) = 0, X Y (no linear correlation), X Y,. : (1) Cov(X, Y ) = Cov(Y, X); (2) Cov(X, Y ) = E(XY ) E(X)E(Y ); (3) Cov(X, X) = Var(X); (4) X Y, Cov(X, Y ) = 0, ; (5) Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z); (6) a, Cov(X, a) = 0; (7) a, b, Cov(aX, by ) = abcov(x, Y ); (8) (X, Y ), Var(X ± Y ) = Var(X) + Var(Y ) ± 2Cov(X, Y ).,, (2), (4) (8). (8). Var(X + Y ), Var(X Y )., Var(X + Y ) = E[(X + Y ) E(X + Y )] 2 = E[(X E(X)) + (Y E(Y ))] 2 = E[(X E(X)) 2 + 2(X E(X))(Y E(Y )) + (Y E(Y )) 2 ] = E[X E(X)] 2 + E[Y E(Y )] 2 + 2E[(X E(X))(Y E(Y ))] = Var(X) + Var(Y ) + 2Cov(X, Y ) (2). Cov(X, Y ) = E[(X E(X))(Y E(Y ))] = E[XY XE(Y ) Y E(X) + E(X)E(Y )] = E(XY ) E(X)E(Y ) E(Y )E(X) + E(X)E(Y ) = E(XY ) E(X)E(Y ) (2), X Y, E(XY ) = E(X)E(Y ), Cov(X, Y ) = 0, (4), X Y, Var(X ± Y ) = Var(X) + Var(Y )..

68 68 R (2), X Y 0 X Y.,, (X, Y ) 1/4 : (1, 0), (0, 1), ( 1, 0), (0, 1), X Y. X Y 0, E(X) = E(Y ) = 0;, (X, Y ) XY 0, E(XY ) = 0. Cov(X, Y ) = E(XY ) E(X)E(Y ) = 0, X Y., X Y., X = 0, Y ; X, Y. X Y,,.,,..,. : (1) 1 ρ XY 1; (2) ρ XY Cov(X, Y ), 0; (3) ρ XY = ±1 X Y, a(a 0) b, Y = ax + b; (4) ρ XY = 1, Y = ax + b a > 0, X Y ; (5) ρ XY = 1, Y = ax + b a < 0, X Y ; (6) ρ XY = 0, Cov(X, Y ) = 0, X Y,,. (7) ρ XY > 0, X Y, ρ XY 1,,, X Y ; (8) ρ XY < 0, X Y, ρ XY 1,,, X Y ; (9) X Y U = ax +b V = cy +d( a, c 0) ρ UV, a, c ρ XY, a, c ρ XY. (1) Schwarz, [Cov(X, Y )] 2 Var(X)Var(Y ) (2.77),. (3),,, P (Y = ax + b) = 1., X Y, 0., P (Y = ax + b) = 1 Y = ax + b.

69 2 69 (6), 0,.,,, X U( 0.5, 0.5), Y = cos X. X Y E(X) = 0, Cov(X, Y ) = E(XY ) E(X)E(Y ) = E(XY ) = E(X cos X) = x cos x dx = 0, ρ XY = 0, X Y,.,,.,. (9).,.,.,,.,.,,.,,.,.., X : X = X E(X) Var(X) (2.78) E(X = 0), Var(X = 1). Y, Cov(X, Y ), ( X E(X) Cov(X, Y ) = Cov = E =, Y E(Y ) ) σ X σ Y ( X E(X) Y E(Y ) σ X σ Y ) E ( X E(X) σ X ) E E[(X E(X))(Y E(Y ))] = Cov(X, Y ) = ρ XY σ X σ Y σ X σ Y ( ) Y E(Y )., [ 1, 1], X Y,,. ( ), (covariance matrix),,.. σ Y

70 70 R 2.8, , n (sufficiently large).,. 1.,?.. A P (A) = p,, M n n A, M n = 1 n n i=1 X i = X 1 + X X n n (2.79) : X i = 1 A ; X i = 0 A. E(X i ) = p., n, M n p., 2.9 ( ) (identically independent distribution) X 1, X 2, µ, σ. ɛ > 0, M n = 1 n n i=1 (law of large numbers, LLN). X i = X 1 + X X n n (2.80) lim P ( M n µ ɛ) = 0 (2.81) n, (weak law of large numbers), (stong law of large numbers), : 2.10 ( ) µ, σ. M n 1 µ, M n = 1 n X 1, X 2, n i=1 (stong law of large numbers). X i = X 1 + X X n n (2.82) ( ) P lim M n = µ = 1 (2.83) n,,., :, ( 1), A P (A).

71 2 71 (law) (theorem),,,.,,,.,, :,.., : X 1, X 2,, X n, S n = X 1 + X X n? (central limit theorem, CLT) :,.,,.,, ( ) X 1, X 2,, X n, E(X i ) = µ, Var(X i ) = σ 2, i = 1, 2,, n. Z n = X 1 + X X n nµ nσ E(Z n ) = 0, Var(Z n ) = 1. Z n, x n, Z n N(0, 1). lim P (Z n x) = 1 x e t2 2 dt n 2π, Z n S n = X 1 + X X n,, n, S n N(nµ, ( ) nσ)., X = S n /n, n, X σ N µ, ;, n.., - (De Moivre Laplace),,, (normal approximation to binomial ditribution) ( ), X i : X 1, X 2,, X n X i 0 1 P (X i ) 1 p p X i. S n = X 1 + X X n, S n n

72 72 R. x, lim P n ( ) S n np x np(1 p) = 1 2π x e t2 2 dt n, S n N[np, np(1 p)]. np np(1 p) B(n, p).,,,,, X i. X i, ,,., S n (X i ),, S n.,,,.,,,,.,,.,,..,.,.,.. Laplace, 20, George Pólya ( , ) 1920 Central Limit Theorem,., n., n 30, n 50.,., n,., n, p. p 0.5,, n ; p 0 1, n. np 10, n(1 p) 10. n 30, n = 100, p = ,. n 30., :,,?,, , 80%,, 95%? X=, a., X B(1 000, 0.8).,

73 2 73 P (X a) 0.95 a.,,.,., = 800 > 10, (1 0.8) = 200 > 10,, X N(800, 12.65), , ( P (X a) = P Z a 800 ) R qnorm(0.95) 1.645, a a a = :, R qnorm(0.95, 800, sqrt(1000*0.8*0.2)) ,,,,., n. µ, X i N(µ, 0.2). X n, 95% µ 0.1,? n, X N n : P (µ 0.1 < X < µ + 0.1) 0.95, P ( X µ < 0.1) P ( X < µ 0.1) 0.05 P Z < µ 0.1 µ n ( ) n P Z < ( µ, 0.2 n ). R qnorm(0.025) , n 1.96 (2.84) 2 n n = 16 95% µ 0.1. n,.

74 74 R 2.33 p, ˆp p. 90% ˆp p 5%,? n. 1, i ; X i = 0, i X i, P (X i = 1) = p, P (X i = 0) = 1 p, i = 1, 2,, n. n n, Y. Y = X i B(n, p), ˆp = Y/n. n, ˆp p. np 10, n(1 p) 10, Y N(np, np(1 p)). ˆp Y, ˆp = Y ) p(1 p) (p, n N n ˆp p, P ( ˆp p 0.05) 0.90 P (ˆp p 0.05) 0.05 P p 0.05 p z p(1 p) 0.05 P 0.05 z p(1 p) 0.05 n R qnorm(0.05) 1.645, 0.05 p(1 p) n n n p(1 p) p(1 p) 0.25(0 < p < 1), n , 271. p.,, : i=1

75 2 75 X 1, X 2,, X n, (µ) (σ), X n, S n n, n, ( (1) X σ N µ, ); n (2) S n N(nµ, nσ); (3) Z n = S n nµ nσ N(0, 1); (3), X i (i = 1, 2,, n), P (X i = 1) = p, P (X i = 0) = 1 p, ˆp = X = S n n n, X = S n = X i, np 10, n(1 p) 10, [ i=1 ] p(1 p) ˆp N p,, X = S n N[np, np(1 p)]. n,. n ( ) X B(2, p), Y B(4, p), P (X 1) = 8/9. P (Y 2). 2. X P (λ), P (X = 1) = P (X = 2), P (X = 4) , 0.10,, 85,. 4., 0.85, 100, 80,,. (1) 0.8. (2) 0.75,? 5.,.,, ;,,. : (1) 3, ; (2) 4 ( 4 ). 6., 2km.,, 5km;,, 20km. 0.6,. 7. X U( 1, 2). 1, x < 0; Y = 1, x 0 Y.

76 76 R 8. B (0,10), x 2 + Bx + 1 = , X( : h), 1 f(x) = 600 e x 600, x > 0; 0, 200h, (, ). 10. X ( : ), 4.,,. 100, 300, , , ,? 12. X N(0, σ). P ( X > k) = 0.1, P (X < k). 13. X N(µ, σ). σ, P ( X µ < σ)? 14. X (1,2), Y = e 2X f Y (y). 15. X 0, x < 0; F X (x) = 1 e x 5, 0 x < 2; 1, x 2 Y = e X, Y. 16. X 0, x < a(a > 0); F (x) = A + B arcsin x, a x < a; a 1, x a (1) A B. (2) X f(x). 17. X Y = 1 + ( 1) X. P (X = n) = 2, n = 1, 2, 3n 18. X U(1, 2), Y = e 2X f Y (y). 19. X Y, E(X) = µ, Var(X) = σ 2. E(X Y ) 2 Var(X Y ).

77 , 3 2. ( : min) N(3, 3), N(1, 2), (0, 1) n,. 22. n,,.,,. ( ). 23. n, 1, 2,, n. k,. : n k=1 k 2 = n 2 = 1 n(n + 1)(2n + 1) X Y λ. U = 2X + Y, V = 2X Y, ρ UV. 25. X Y ρ. U = ax + b V = cy + d, a, c n. H T Head Tail. H T. 27., 10 min,. (1) h 15 h ; (2) 95%, 15 h? kw %, kw, 99%?( ) 29. p, n, m, n m/n p %. 30.., , (critical value),., a 5%,.,? 31. X F (x) f(x). X. (hazard rate) (failure rate) λ(t), : λ(t) = f(t), t 0 (2.85) 1 F (t)

78 78 R t., t, dt P [X (t, t + dt) X > t]., P [X (t, t + dt), X > t] P [X (t, t + dt) X > t] = P (X > t) P [X (t, t + dt)] = P (X > t) f(t) dt 1 F (t) : P [X (t, t + dt), X > t] X (t, t + dt) X > t., λ(t) t (conditional probability intensity). (1), X., X X.. (2) X Exp(λ)., t,., λ(t).. (3) : (death rate).,.,?. (4) t(t 40), λ(t) = (t 40) , t , ( 2.9). ( 2.10). R.

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