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MECS 07. Statistical Models

# MECS 07. Statistical Models - Modeling and Evaluation of...

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Unformatted text preview: : Modeling and Evaluation of Computer Systems (MECS) :7 (Statistical Models) : (Mohammad Abdollahi Azgomi) [email protected] :7 MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 2 1 (probabilistic) . (deterministic) . : (sampling) . . ( ) (K-S K2 ) (goodness of fit) . . MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 3 : (discrete random variables) (continuous random variables) (cumulative distribution function) (expectation) MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 4 2 (finite) X X . (countably finite) () : . X Rx={0, 1, 2, ...} p(xi) = P(X = xi) : Rx X (range space) : p(xi) xi : i = 1, 2, ... p(xi) 1. p( xi ) 0, for all i 2. i =1 p( xi ) = 1 i=1, 2, ... [xi, p(xi)] p(xi) (probability distribution) . X (pmf: probability mass function) 5 MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE () : . : MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 6 3 (Rx) X . () ... : [a, b] X P(a X b) = f (x)dx a b (pdf: probability density function) f(x) . MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 7 : f(x) 1. f ( x ) 0 , for all x in R X 2. RX f ( x ) dx = 1 3. f ( x ) = 0 , if x is not in R X : 1. P ( X = x0 ) = 0, because f ( x)dx = 0 x0 x0 2. P (a X b) = P (a < X b) = P (a X < b) = P (a < X < b) MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 8 4 X : : pdf 1 -x/2 e , x0 f ( x) = 2 0, otherwise . 2 X : 3 2 P(2 X 3) = 3 2 1 -x/ 2 1 3 e dx = e - x / 2 dx = 0.145 2 2 2 9 MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE X (cumulative distribution function (cdf)) : Fx F(X) F(x) = Fx = P(X x) F ( x) = p( xi ) all xi x : X : X F ( x) = f (t )dt - x : 1. F is nondecreasing function : If a < b, then F (a ) F (b) 2. lim x F ( x) = 1 3. lim x - F ( x) = 0 : . cdf X P( a < X b) = F (b) - F (a ), for all a < b MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 10 5 : F ( x) = 1 x -t / 2 -x / 2 0 e dt = 1- e 2 P(0 X 2) = F (2) - F (0) = F (2) = 1- e-1 = 0.632 P( 2 X 3) = F (3) - F (2) = (1 - e - (3 / 2 ) ) - (1 - e -1 ) = 0.145 MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 11 : (expected value) E ( x) = xi p ( xi ) all i : X : X E ( x) = xf ( x)dx - Var(X) V(X) 2 : V(X) = E[(X E[X]2)] V(X) = E(X2) [E(x)]2 . (standard deviation) MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 12 6 : 1 -x / 2 E( X ) = xe-x / 2dx = - xe + e-x / 2dx = 2 0 2 0 0 : E[X2] E( X 2 ) = 1 2 -x / 2 -x / 2 + e-x / 2dx = 8 0 x e dx = -x 2 e 0 2 0 : V ( X ) = 8 - 22 = 4 = V (X ) = 2 MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 13 : (queueing systems) (inventory and supply-chain systems) (reliability and maintainability) :(limited data) . MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 14 7 . : : : (Weibull) (gamma) . MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 15 : .(lead time) : : . MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 16 8 :(time to failure (TTF)) : (standby redundancy) : TT TTF : : MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 17 : . : (Bernoulli) (binomial) .(hyperexponential) MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 18 9 . : MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 19 n : : . j Xj = 1 . j Xj = 0 : ( ) x j = 1, j = 1,2,...,n p, p j ( x j ) = p( x j ) = 1- p = q, x j = 0,j = 1,2,...,n 0, otherwise E(Xj) = p V(Xj) = p(1-p) = pq : : n : p(x1,x2,..., xn) = p1(x1)p2(x2) ... pn(xn) MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 20 10 n X : n x n-x p q , x = 0,1,2,...,n p( x) = x 0, otherwise . E(x) = p + p + ... + p = n*p V(X) = pq + pq + ... + pq = n*pq MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE x n-x : 21 : (X) : q x-1 p, x = 0,1,2,...,n p( x) = otherwise 0, E(x) = 1/p V(X) = q/p2 : . k (X) : k p Y y -1 y-k k q p , y = k, k +1, k + 2,... p( x) = k -1 0, otherwise E(Y) = k/p V(X) = kq/p2 22 MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 11 . : cdf pdf > 0 e - x p( x) = x! , x = 0,1,... 0, otherwise E(X) = = V(X) e- i F ( x) = i! i =0 x MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 23 : . = 2 p(3) p(3) = e-223/3! = 0.18 : F(x) = F(3) F(2) = 0.857-0.677 = 0.18 = 1 p(0) p(1) = 1 F(1) = 0.594 p(2 or more) MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 24 12 ) .( : (uniform) (exponential) (normal) (Weibull) (lognormal) MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 25 U(a, b) (a, b) X : cdf pdf 1 , a xb f ( x) = b - a 0, otherwise x<a 0, x -a F ( x) = , a x<b b - a xb 1, : . [F(x2) F(x1) = (x2-x1)/(b-a)] P(x1 < X < x2) V(X) = (b-a)2/12 : E(X) = (a+b)/2 : (random numbers) U(0, 1) . MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 26 13 >0 X : e - x , x 0 f ( x) = 0, x < 0 : CDF F ( x) = e -x dx = 1 - e -x , x 0 0 x 8 . MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 27 : pdf X f ( x) = 1 x - 2 1 exp - , - < x < 2 2 - < < : : >0 2 . X ~ N(,2) : lim x - f ( x ) = 0, and lim x f ( x) = 0 :f(x) f(-x)=f(+x) : pdf . x = pdf MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 28 14 : pdf X x - -1 x - exp - , x f ( x) = 0, otherwise : (- < < ) :(location) ( > 0) :(scale) ( > 0) :(shape) When = 1, X ~ exp( = 1/) = 1 = 0 : MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 29 pdf (lognormal) X : =1, 2=0.5,1,2. 1 (ln x - ) 2 exp - , x > 0 f ( x ) = 2 x 2 2 0, otherwise E(X) = e+ /2 : V(X) = e2+ /2 (e - 1) : 2 2 2 : (Y = ln X ) X = eY ~ lognormal(, 2) : Y ~ N(, 2) . 2 MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 30 15 (observed values) . (empirical distribution) . (extrapolation) (interpolation) . : . . : MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 31 . (input modeling) . . MECS#7: Statistical Models - By: M. Abdollahi Azgomi - IUST-CE 32 16 ...
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