Στατιστικές...
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. . . . MSc (..../ ) Team Site: A.E.A.C. Co. Project Manager-Site Administrator e-mail: [email protected] , My Blog. 16/07/2011
. . / . . . . . 1) 2) ( , , ), . 3) . 4) / .
. . 1) 2) . ) ) 3) . . [,]. . 1) (0,1,2,3, ), 2) (0%, 10%, 30%, 40%). . , 2.5 . ( ) 10.78%. . [,]. P ( a < x < b) = f ( x)dxa b
. 1) (15, 50, 60 ), 2) (1.75, 1.80, 1.90 ). . , 20 2 , 35 5 4 . 1.76, 1.77, 1.80 4 ( ). (-,+). (). [,] ( ). x> x->0, .
Easy Fit 5.1 excel. Easy Fit 5.1 50 , . , .
. 4 . Easy Fit 5.1 .
Easy Fit 5.1. : Beta, Johnson SB, Kumaraswamy, Pert, Power Function, Reciprocal, Triangular, Uniform, : Cauchy, Error, Gumbel Max Gumbel Min, Hyperbolic Secant, Johnson SU, Laplace (Double Exponential), Logistic, Normal, t-Student Burr, Levy, Gamma, Inverse Gaussian, F Distribution, Fatigue Life (Birnbaum-Saunders), Frechet, Chi-Squared, Dagum, Erlang, Exponential, Weibull, Rice, Rayleigh, Pearson, Pareto, Nakagami, Lognormal, Log-Logistic, Log-Gamma. : Bernoulli, Binomial Discrete Uniform, Geometric, Hypergeometric, Logarithmic, Negative Binomial, Poisson. : Generalized Extreme Value, Generalized Logistic, Generalized Pareto, Phased Bi-Exponential, Phased Bi-Weibull, Wakeby.
. . , , .
----------------
) [ (-, +)]
(Laplace-Gauss NORMAL DISTRIBUTION) . Gauss . , , ( =3) ( =0) , .
1 (...)
f ( x) =
1
* 2
1*( x ) 2 e 2 ,
~(,)
1
- Probability Density Function - Cumulative Distribution Function
12 t x * e 20
F(x)=
1 x , Laplace // (x)= 2
dt
F(x)=z=
x
,
>0 (scale parameter/ ),
R (location parameter/ ) NORMAL
< x < +
Probability Density Function0,32
0,28
0,24 0,2
f(x)
0,16
0,12 0,08
0,04
0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
xHis togram Norm al
~ (,2) [-, +] 68.28% [-2, +2] 95.44% [-3, +3] 99.75%
1) ~(8,9). P (5 X 10) . =8 2=9
z=
x
z= z=
58 = -1 3 10 8 = 0.66 3
P (5 X 10) = P (1 z 0,66) = P ( z 0,66) - P ( z 1) = P ( z 0,66) - (1 P ( z 1))
0,7454-(1-0,8413) = 0,8413 (. )
, (-, 0.66) (-, -1). (-1, 0.66)
2) ~(3,2) P ( X 1,5) = 0,7291. 2
z=
x
z=
1,5 3
z=
1,5
P ( X 1,5) = 0,7291.)
z= 0.61 (.
z 0), z= -0.61 -0.61=
1,5
=2,46 ( 0 (scale parameter), R (location parameter) < x < +
: . 2 .
: , ( ), ( )
>3)
2
( 0 (scale parameter ) , R (location parameter) arctan < x < +
CAUCHY
Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
xHistogram Cauchy
JOHNSON SU (JOHNSON SU DISTRIBUTION)
Norman Lloyd Johnson (1917-2004). Johnson SB Log-Normal . , , . , , ( =3) ( =0). , , . , Johnson SB, Johnson, SU Log-Normal .
f ( x) =
* 2 * z 2 + 1
*e
2 + 1)) 2 0.5 * ( + * ln( z + z
,
~ Jsu(,,,)
F ( x) = ( + * ln( z + z 2 + 1))
t x 1 x 2 z = , Laplace // (x)= *e dt
2
2
0
,,, ,>0. x (,+) , (shape parameter), (scale parameter), (location parameter)
JOHNSON SU
Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
xHistogram Johnson SU
LAPLACE DOUBLE EXPONENTIAL (LAPLACE DOUBLE EXPONENTIAL DISTRIBUTION)
Pierre-Simon Laplace. double exponential ( ) .. Laplace ( ).
* | x |
f ( x) = * e2
,
~ Laplace(,)
F ( x) =
1 * ( x) *e 2 1 * (x ) 1 *e 2
x
x>
, >0. x (,+) (scale parameter ) , (location parameter)
LAPLACE
Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
xHistogram Laplace
LOG-NORNAL 3- (LOG-NORNAL DISTRIBUTION 3)
Log-Normal . Log-Normal . 3 (, ) Log-Normal . ( / , , , ) )
3
: Log-Normal
f ( x) =
ln( x ) 0.5 * e
2, ~ LogN(,,)
( x ) * * 2 *
F ( x) =
ln( x )
t 1 Laplace // (x)= * e 2 dtx
2
2
0
, , >0. x ( ,+) (scale parameter ), (shape parameter), (location parameter)
: =0, Log-Normal 2- (Log-Normal 2). Log-Normal 2 =0
LOG-NORMAL
Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
x
Histogram
Lognormal (3P)
GAMMA 3- (GAMMA 3 DISTRIBUTION)
(waiting time models). , , Gamma .
(x )
f ( x) =
(x )
1
* ( )
*e
,
~ (,,)
F ( x) =
( x ) / ( ) ( )
x()
Gamma// x() = t a 1 * e t dt0
x
, , ,>0, R
x ( ,+)
(shape parameter), (scale parameter) , (location parameter)
: =0, Gamma 2- (Gamma 2). Gamma 2 =0 Gamma Erlang
GAMMA Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
x
Histogram
Gamma (3P)
WEIBULL 3- (WEIBULL 3 DISTRIBUTION)
Waloddi Weibull (1887-1979), . (Exponential) Rayleigh . , (=1) Rayleigh (=2). Weibull . 1 . (. ) Weibull , , ..
f ( x) =
x *
1
x *e
,
~ W(,,)
x F ( x) = 1 e
, , ,>0, R
x ( ,+)
(shape parameter), (scale parameter) , (location parameter)
: =0, Weibull 2- (Weibull 2). Weibull 2 =0
WEIBULL
Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
x
Histogram
Weibull (3P)
FATIGUE LIFE (BirnbaumSaunders) 3- (FATIGUE LIFE 3 DISTRIBUTION)
. , . Lognormal, Exponential and Weibull.
, , , .
x f ( x) =
1 x x * * 2* a *(x ) x
+
~ BS(,,)
1 x F ( x) = *
x 2 x
x2 e 2
t 1 Laplace // (x)= * e 2 dt
2
0
2 *
, , ,>0, R
x ( ,+)
(shape parameter), (scale parameter) , (location parameter)
: =0,
Fatigue Life
2- (Fatigue Life 2). Fatigue Life 2 =0
FATIGUE LIFE
Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
x
Histogram
Fatigue Life (3P)
ERLANG 3 (ERLANG 3 DISTRIBUTION)
Erlang Agner Krarup Erlang (1878 1929). Gamma Exponential . Agner Krarup Erlang . .
Erlang Gamma shape parameter Erlang (m) shape parameter Gamma () . Erlang Gamma. m=1 Erlang Exponential .
(x )
f ( x) =
(x ) *e m * ( m)
m 1
,
~ Erlang(m,,)
F ( x) =
( x ) / ( m) ( m)
x()
Gamma// x() = t a 1 * e t dt0
x
m, , m N * , >0, R
x ( ,+)
m (shape parameter), (scale parameter) , (location parameter)
: =0, Erlang 2 (Erlang 2). Erlang 2 =0
ERLANG
Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
x
Histogram
Erlang (3P)
(EXPONENTIAL DISTRIBUTION)
Poisson . (.. ), , , , .. . , .
f ( x) = * exp *( x )
,
~ Exp(,)
F ( x) = 1 exp * ( x )
, >0, R
x ( ,+)
(scale parameter) , (location parameter)
Probability Density Function0,32 0,28 0,24 0,2
f(x)
0,16 0,12 0,08 0,04 0 -0,06 -0,04 -0,02 0 0,02 0,04 0,06
x
Histogram
Exponential (2P)
PEARSON TYPE 6 4- (PEARSON TYPE 6 4 DISTRIBUTION)
Karl Pearson (1857-1936) . , , , , . . Pearson Type6 5 .