Probability theory STATISTICAL METHODS FOR SAFETY ANALYSIS FMS065 TABLE OF FORMULÆ (2016) Basic probability theory. One-dimensional random variables

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1 Lund University Centre for Mthemticl Sciences Mthemticl Sttistics STATISTICAL METHODS FOR SAFETY ANALYSIS FMS065 TABLE OF FORMULÆ (06) Probbility theory Bsic probbility theory Let S be smple spce, nd let P be probbility on S. Then, for ll events A, B, A, A,..., A n S, () Kolmogorov s xioms (.) 0 P(A) (.) P(S) = (.3) P(A B) = P(A) + P(B), if A nd B re disjoint. () P(A B) = P(A) + P(B) P(A B) (3) A nd B re independent P(A B) = P(A) P(B). (4) Conditionl probbility: P(B A) = (5) Lw of totl probbility: P(B) = P(A B). P(A) P(B A i ) P(A i ), whenever A,..., A n re pirwise disjoint nd stisfy (6) Byes theorem: P(A i B) = P(B A i) P(A i ) P(B) whenever A,..., A n re pirwise disjoint nd stisfy One-dimensionl rndom vribles = n A i = S. P(B A i ) P(A i ) n j= P(B A j) P(A j ), n A k = S. (7) Distribution function for the rndom vrible X : F X (x) = P(X x). (8) Probbility-mss function for the discrete rndom vrible X : p X (x) = P(X = x). k= (9) Density function for the continuous rndom vrible X : f X (x) = df X (x) dx differentible. for ll x where F X is (0) If X is discrete, then P( < X b) = F X (b) F X () = p X (x) x ],b] If there is no element x in ], b] such tht p X (x) 0, then F X (b) F X () = 0.

2 ii TABLE OF FORMULÆ, FMS065, 06 () If X is continuous, then P( < X b) = F X (b) F X () = Two-dimensionl rndom vribles b f X (x) dx () oint distribution function for the two rndom vribles X nd Y : F X,Y (x, y) = P(X x Y y) (3) oint probbility-mss function for the two discrete rndom vribles X nd Y : p X,Y (x, y) = P(X = x Y = y) (4) oint density function for the two continuous rndom vribles X nd Y : f X,Y (x, y) = F X,Y (x, y) x y for ll (x, y) where the derivtive exists. (5) If X nd Y both re discrete: P ( (X, Y ) A ) = (x,y) A p X,Y (x, y) If there is no pir (x, y) in A such tht p X (x) 0, then P ( (X, Y ) A ) = 0. (6) If X nd Y both re continuous: P ( (X, Y ) A ) = f X,Y (x, y) d(x, y) Conditionl distributions (x,y) A (7) Conditionl distribution function: F X Y (x y) = P(X x Y = y). (8) Conditionl probbility-mss function for the discrete rndom vrible X : p X Y (x y) = P(X = x Y = y) p X,Y (x, y), p Y (y) 0 If Y is lso discrete, then p X Y (x y) = p Y (y) 0, p Y (y) = 0. (9) Conditionl density function for the continuous rndom vrible X : f X Y (x y) = F X Y (x y) x f X,Y (x, y), f Y (y) 0 If Y is lso continuous, then f X Y (x y) = f Y (y) 0, f Y (y) = 0.

3 TABLE OF FORMULÆ, FMS065, 06 iii (0) Byes theorem: p X Y (x y) p Y (y), p X (x) 0, (0.) X discrete nd Y discrete: p Y X (y x) = p X (x) 0, p X (x) = 0. (0.) Y continuous: replce p Y X (y x) by f Y X (y x) nd p Y (y) by f Y (y). (0.3) X continuous: replce p X Y (x y) by f X Y (x y) nd p X (x) by f X (x). () Mrginl probbility-mss function for the discrete rndom vrible X : (.) Y is discrete: p X (x) = y p X Y (x y) p Y (y) = y p X,Y (x, y) (.) Y is continuous: p X (x) = p X Y (x y) f Y (y) dy () Mrginl density function for the continuous rndom vrible X : (.) Y is discrete: f X (x) = f X Y (x y) p Y (y) (.) Y is continuous: f X (x) = y; p Y (y) 0 f X Y (x y) f Y (y) dy = (3) If X nd Y re independent, (3.) then F X,Y (x, y) = F X (x) F Y (y), (3.) then p X,Y (x, y) = p X (x) p Y (y) if X nd Y re discrete, (3.3) then f X,Y (x, y) = f X (x) f Y (y) if X nd Y re continuous, (3.4) then F X Y (x y) = F X (x), (3.5) then p X Y (x y) = p X (x) if X is discrete, (3.6) then f X Y (x y) = f X (x) if X is continuous. Lw of totl probbility gin Lw of totl probbility: Let A be n event. f X,Y (x, y) dy (4) If X is discrete rndom vrible, then P(A) = x P(A X = x) p X (x). (5) If X is continuous rndom vrible, then P(A) = Expecttion, vrince, nd the like P(A X = x) f X (x) dx. (6) Let g be rel-vlued function x g(x). Then the expecttion of g(x ) is given by (6.) E ( g(x ) ) = g(x) p X (x), if X is discrete, (6.) E ( g(x ) ) = p X (x) 0 g(x) f X (x) dx, if X is continuous.

4 iv TABLE OF FORMULÆ, FMS065, 06 (7) Let g be rel-vlued function (x, y) g(x, y). Then the expecttion of g(x, Y ) is given by (7.) E ( g(x, Y ) ) = x,y g(x, y) p X,Y (x, y), if X nd Y re discrete, (7.) E ( g(x, Y ) ) = g(x, y) f X,Y (x, y) d(x, y), ( (X ) ) (8) Vrince: V(X ) = E E(X ) = E(X ) ( E(X ) ). (9) Stndrd devition: D(X ) = V(X ). (30) Coefficient of vrition: R(X ) = D(X )/E(X ). if X nd Y re continuous. (3) Covrince: C(X ; Y ) = E( (X E(X ) )( Y E(Y ) ) ) = E(XY ) E(X ) E(Y ). (3) C(X, X ) = V(X ). (33) Coefficient of correltion: r(x, Y ) = C(X, Y ) D(X ) D(Y ). (34) Expecttion is liner, i.e. E(X + by + c) = E(X ) + be(y ) + c. (35) V(X ± by ± c) = V(X ) + b V(Y ) ± b C(X, Y ) (36) Covrince is biliner, i.e. C(X ± by, cz) = cc(x, Z) ± bcc(y, Z). (37) For independent rndom vribles X, Y : E(XY ) = E(X ) E(Y ). (38) Guss pproximtions: Let g be rel-vlued function (x, x,..., x n ) g(x, x,..., x n ). Then E(g(X,..., X n )) g(e(x ),..., E(X n )). V(g(X,..., X n )) ci V(X i ) + c i c j C(X i, X j ), i n j n i<j where c i = g x i (E(X ),..., E(X n )) = g x i (E(X ),..., E(X n )). Norml (Gussin) distribution (39) Univrite norml (Gussin) distribution (s > 0): X N(m, s ) X m s N(0, ) (40) Bivrite norml (Gussin) distribution: Let m, m, s, s, nd r be rel numbers (s > 0, s > 0, < r < ). If (X, Y ) N(m, m, s, s, r), then (40.) f X,Y (x, y) = ps s r e (x m ) s + (y m ) s r x m s y m s «,

5 TABLE OF FORMULÆ, FMS065, 06 v (40.) X N(m, s ), Y N(m, s ), C(X, Y ) = rs s, r(x, Y ) = r, ( ( ) ) (40.3) f X Y (x y) = ps r e s x m ( r +r s ) s (y m ), i.e. N ( m + r s s (y m ), s ( r ) ) (40.4) X + by N(m + bm, s + b s + b r s s ) for ll rel numbers nd b. Limit theorems (4) Lw of Lrge Numbers (LLN): Let X, X,... be independent nd identiclly distributed rndom vribles with existing expecttion E(X i ) = m. Then Y n = X X n n E(X i ), when n. (4) Centrl Limit Theorem: Let X, X,..., X n be independent nd identiclly distributed rndom vribles with existing expecttion E(X i ) nd existing stndrd devition D(X i ) = s <. Then Y n = X X n AsN(n m, n s ), when n. (43) We hve pproximtely (43.) Bin(n, p) Po(np) if p 0. nd n 0. (43.) Bin(n, p) N(np, np( p)) if np( p) 0. (43.3) Po(m) N(m, m) if m 5. Sums of rndom vribles (44) Let X N(m, s ),..., X n N(m n, s n ) be n independent, normlly distributed rndom vribles. For ny set c,..., c n of n rel numbers, we hve ( c i X i N c i m i, c i si ). (45) If X nd X re independent, then (45.) X Bin(n, p), X Bin(n, p) X + X Bin(n + n, p) (45.) X Po(m ), X Po(m ) X + X Po(m + m ). (45.3) X Gmm(, b), X Gmm(, b) X + X Gmm( +, b). (45.4) X q (f ), X q (f ) X + X q (f + f )

6 vi TABLE OF FORMULÆ, FMS065, 06 Sttistics Point estimtion Let x,...,x n be observtions of n independent, identiclly distributed rndom vribles with expecttion m nd stndrd devition s. Then unbised estimtions of m nd s re given by (46) m = n x i = x (47) (s ) = n (x i m), m known. (48) (s ) = s = n Confidence intervls (x i x), m unknown. (49) Let j be some prmeter, nd let (r.v.) be n estimtor of j such tht is (pproximtely) normlly distributed with expecttion j. Let j be the estimte of j, i.e. let j be the observtion of. Then I j = [j l ( g)/ d( ); j + l ( g)/ d( )] I j = [j l g d( ); ] I j = [; j + l g d( )] (two-sided), (one-sided, bounded below), (one-sided, bounded bove) re confidence intervls for j with pproximtive confidence level g (g is typiclly lrge, g = 0.95, g = 0.99,... ). Here, d( ) is the stndrd error of the estimtor. d( ) = Hypothesis testing D( ) if D( ) is known nd independent of j, ( D( ) ) if D( ) is unknown or dependent on j. (50) Let j be prmeter. We wnt to test the simple hypothesis H 0 : j = j 0 on the significnce level ( is typiclly smll, = 0.05, = 0.0,... ). Then the test will be Reject H0 j 0 I j Do not reject H 0 j 0 I j where I j is confidence intervl of j with (pproximtive) confidence level. If H : j j 0, then choose I j to be two-sided. If H : j > j 0, then choose I j to be one-sided nd bounded below. If H : j < j 0, then choose I j to be one-sided nd bounded bove.

7 TABLE OF FORMULÆ, FMS065, 06 vii (5) The q test. Let H 0 be hypothesis bout the expressed by probbilities p,..., p r. We hve n observtions. Clculte r (x i np i ) Q =. np i Reject H 0 if Q > q (r ). Byesin updting (5) Let be the prmeter modelled s rndom vrible. Let X be the observed rndom vrible with observed vlue x. (5.) is discrete, X is discrete: p post (j) = c P(X = x = j) pprior (j), where c = P(X = x) = j P(X = x = j) p prior (j). (5.) is continuous, X is discrete: f post prior (j) = c P(X = x = j) f (j), where c = P(X = x) = P(X = x = j) f prior (j) dj. (5.3) If X is continuous: replce P(X = x = j) by f X (x j) nd P(X = x) by f X (x). (53) Prticulrly, Prior distribution of Conditionl distribution of X, given = j Posterior distribution of (x is the observtion of X ) (53.) Gmm(, b) Po(j t) Gmm( + x, b + t) (53.) Bet(, b) Bin(n, j) Bet( + x, b + n x) (54) If the event A nd the rndom vrible X re independent on condition tht is known, then we hve the lw of totl probbility: (54.) P post (A) = (j), if is discrete, (54.) P post (A) = p post (j) 0 P(A = j) p post P(A = j) f post (j) dj, if is continuous. (55) If we hve observed the occurrence of n event B (insted of hving observed the rndom vrible X ), then: (55.) p post (j) = P(B = j) pprior(j), if is discrete, (55.) f post prior (j) = P(B = j) f (j), if is continuous. If the event A nd the event B re independent on condition tht is known, then (54.) nd (54.) still re vlid.

8 viii TABLE OF FORMULÆ, FMS065, 06 Miscellneous The Poisson process Let N (t) be the number of events tking plce in the time intervl ]0, t]. If N (t) is Poisson process with constnt intensity l, then (56) N (t) Po(lt). (57) Time lgs between consecutive events re independent nd exponentilly distributed with expecttion /l. (58) The number of events occurring in time intervl I nd the number of events occurring in nother time intervl I re independent if I nd I re disjoint. Filure rte Let T be positive, continuous rndom vrible with density function f T nd distribution function F T. (59) l(t) = f T (t) F T (t), t 0 nd F T (t). (60) P(T > t) = exp ( t ) l(s) ds, t 0. 0 (6) P(t < T t + D T > t) l(t) D, if D is smll (D > 0). Quntiles Quntiles is the sme s frctiles. Let be rel number such tht 0 < <. Let X be continuous rndom vrible with distribution function F X. (6) The -quntile (denoted x ) is defined to be ny number such tht P(X > x ) = or, equivlently, F X (x ) =. (63) x /4 = x 0.5 is clled the upper (distribution) qurtile. x / = x 0.5 is clled the (distribution) medin. x 3/4 = x 0.75 is clled the lower (distribution) qurtile. x 0.0, x 0.0,..., x 0.98, x 0.99 re clled the (distribution) percentiles. (64) Quntiles l bsed on the stndrd-norml distribution re denoted l : If X N(0; ), then P(X > l ) = F(l ) = l = F ( ), where F (...) is the inverse function of F. (So, F (...) hs nothing to do with (64.) l = l (for ll such tht 0 < < ) (64.) l for some vlues of re found in (99) F(...).)

9 TABLE OF FORMULÆ, FMS065, 06 ix Cornell s relibility index Let h(r,..., R k, S,..., S n ) be the filure function of k rndom strength vribles R,..., R k nd n rndom lod vribles S,..., S n. Let E ( h(r,..., R k, S,..., S n ) ) > 0. (65) Cornell s sfety index b C is defined s b C = E( h(r,..., R k, S,..., S n ) ) D ( h(r,..., R k, S,..., S n ) ). (66) If h(r,..., R k, S,..., S n ) is normlly distributed, then the probbility P f of filure is P f = P ( h(r,..., R k, S,..., S n ) 0 ) = F(b C ). (67) An upper bound for the probbility P f of filure is P f = P ( h(r,..., R k, S,..., S n ) 0 ) Log-norml distribution + b C (68) Let X be log-normlly distributed rndom vrible, i.e. ln X N(m, s ) or ln( X ) x / N(0, s ). Then the coefficient of vrition is given by D(X ) E(X ) = e s (69) Let X nd X be two independent rndom vribles. Then ln X N(m, s ), ln X N(m, s ) ln(x k X k ) N(k m +k m, k s +k s ) Mximum nd minimum Let X,..., X n be n independent, identiclly distributed rndom vribles with distribution function F X (x). If we define then X mx = mx(x,..., X n ) nd X min = min(x,..., X n ), (70) F Xmx (z) = ( F X (z) ) n, (7) F Xmin (z) = ( F X (z) ) n. Some functions (7) The gmm function is defined (for p > 0) by (7.) G(p) = 0 x p e x dx, p > 0 Some properties of the gmm function: (7.) G(p) = (p )!, p ; ; 3;...} (7.3) G( ) = p (7.4) G(p + ) = p G(p); p > 0

10 x TABLE OF FORMULÆ, FMS065, 06 (73) The incomplete gmm function is defined (for p > 0, x 0) by (73.) G(p, x) = x x p e x dx, x 0, p > 0 A property of the incomplete gmm function: (73.) G(p, 0) = G(p), p > 0 (74) The incomplete bet function is defined (for > 0, b > 0, 0 x ) by (74.) B(x,, b) = x 0 A property of the incomplete bet function: x ( x) b dx, 0 x, > 0, b > 0 (74.) B(,, b) = G() G(b) G( + b), > 0, b > 0 Tble of distributions Distribution (75) Hypergeometric distribution (76) Binomil Bin(n, p) (77) Poisson Po(m) ( N )( N ) x n x ( p(x) = N+N ), x Z [n min(n,n), min(n,n )] n 0, otherwise ( ) n p x ( p) n x, x = 0,,..., n p(x) = x 0, otherwise m mx e, x = 0,,... p(x) = x! 0, otherwise Prmeter restrictions N =,,... N =,,... n = =,..., N +N n =,,... 0 < p < Expecttion n N Vrince N + N N +N n N +N n N N (N +N ) np np( p) m > 0 m m (78) Geometric Ge(p) p(x) = p( p) x, x = 0,,... 0, otherwise 0 < p < p p p p (79) First success distribution (80) Uniform U(, b) p(x) = p( p) x, x =,,... 0, otherwise f (x) = b, < x < b 0, otherwise 0, x x b, < x < b, x b 0 < p < < b p + b p p ( b)

11 TABLE OF FORMULÆ, FMS065, 06 xi Distribution (8) Bet Bet(, b) Γ( + b) f (x) = Γ() Γ(b) x ( x) b, 0 < x < 0, otherwise 0, x 0 Γ( + b) B(x,, b), 0 < x < Γ() Γ(b), x Prmeter restrictions > 0, b > 0 Expecttion Vrince + b b ( + b) ( + b + ) (8) Norml (Gussin) distribution, N(m, σ ) f (x) = (x m) e s πσ ( ) σ > 0 m σ x m Φ σ (83) Log-norml distribution, ln X N(m, σ ) 0, x 0 Φ( ln x m ), x > 0 σ σ > 0 e m+s / e m+s e m+s (84) Log-norml 0, ( x 0 distribution, Φ ln X x / N(0, σ ) σ ln x ), x > 0 x / (85) Gmm Gmm(, b) 0, x 0 f (x) = b Γ() x e b x, x 0 0, x 0 Γ(, b x), x > 0 Γ() x / > 0, σ > 0 x / e s / x / (e s e s ) > 0, b > 0 b b (86) Exponentil Exp() 0, x 0 e x/, x > 0 > 0 (87) Gumbel (type I extreme vlue) distribution e e (x b)/ > 0 b + γ π 6 (88) Fréchet (type II extreme vlue) distribution 0, x b e x b c, x > b (89) Type III e x b c, x < b extreme vlue, x b distribution > 0, c > 0 > 0, c > 0 b + Γ( /c) [ Γ( c ) ( Γ( c ) ) ] b Γ( + /c) [ Γ( + c ) ( Γ( + c ) ) ] ) Φ(x) is tbulted in (98). ) Here, x / denotes the distribution medin of the rndom vrible X. ) γ is Euler s constnt. γ = lim n ( ( n k= k ) ln n) = ) Expecttion exists if nd only if c >. Vrince exists if nd only if c >. ) If X is type III extreme vlue distributed rndom vrible, then X (i.e. the negtive of X ) is Weibull distributed. Therefore the type III extreme vlue distribution now nd then is clled the extreme vlue distribution of Weibull type.

12 xii TABLE OF FORMULÆ, FMS065, 06 Distribution Prmeter restrictions Expecttion Vrince (90) Weibull distribution 0, x b e x b c, x > b > 0, c > 0 b + Γ( + /c) [ Γ( + c ) ( Γ( + c ) ) ] (9) Ryleigh distribution (9) Chi-squre χ (n), Gmm( n, ) 0, x b e x b, x > b 0, x 0 f (x) = / Γ( n ) (x/)(n/) e x/, x > 0 0, x 0 Γ( n, x ) Γ( n ), x > 0 > 0 b + π ( π 4 ) n =,,... n n (93) Student s t-distribution, t(n) (94) Fisher s F-distribution, F(n, n ) (95) Preto (c > 0) f (x) = n Γ( n+ ) π Γ( n ) ( + x n )(n+)/ B( n n+x, n, ) B(, n, ), x < 0 B( n n+x, n, ) B(, n, ) ; x 0 [x 0] = 0 f (x) = [x > 0] = n+n Γ( ) = Γ( n n ) Γ( ) n n/ n n/ x n (n + n x) (n+n)/ 0; x 0 B( n n, n +n x, n ) B(, n, n ), x > 0 0, x 0 ( c x )/c, 0 < x < c, x c n =,,... 0 n =,,... n =,,... > 0, c > 0 n n c + n n n (n + n ) n (n ) (n 4) (c + )(c + ) (96) Preto distribution, (c < 0) 0, x 0 ( + c x ) / c, x > 0 > 0, c < 0 c + (c + )(c + ) (97) Preto distribution, (c = 0) 0, x 0 e x/, x > 0 > 0 ) Vrince exists if nd only if n 3. ) Expecttion exists if nd only if n 3. Vrince exists if nd only if n 5. ) Expecttion exists if nd only if c >. Vrince exists if nd only if c >. ) This is n exponentil Exp().

13 TABLE OF FORMULÆ, FMS065, 06 xiii Tble of the stndrd-norml distribution function (98) If X N(0; ), then P(X x) = F(x), where F( ) is non-elementry function given by x p e u du. This tble gives the function vlues F(x) for x = 0.00, 0.0,..., For negtive vlues of x, use tht F( x) = F(x) Tble of some quntiles of the stndrd norml distribution (99) l = F ( ), l = l α λ

14 xiv TABLE OF FORMULÆ, FMS065, 06 Tble of quntiles of Student s t-distribution (00) If X t(n), then the -quntile t (n) is defined by P ( X > t (n) ) =, 0 < < This tble gives the -quntile t (n) for = 0., 0.05, 0.05, 0.0, 0.005, 0.00, nd for n =,,..., 9, 30, 40, 60, 0. For vlues of 0.9, use tht t (n) = t (n), 0 < < n

15 TABLE OF FORMULÆ, FMS065, 06 xv Tble of quntiles of the q distribution (0) If X q (n), then the -quntile q (n) is defined by P ( X > q (n) ) =, 0 < < This tble gives the -quntile q (n) for = , 0.999, 0.99, 0.975, 0.95, 0.05, 0.05, 0.0, 0.005, 0.00, nd for n =,,..., 9, 30, 40, 50,..., 90, 00. n α This is version