8
25
,
a full moon day;Distribution
Last updated on
2010
8 25,
a full moon day;
WHEN random number(s) is/are needed, random variable(s) must be calculated, and then WHAT random variable x will be defined into WHICH parameter of pressure machine ... ; Approx. 22 distributions are available to read;
Develop, design, and engineer a new 2,3 dimensional differentiable manifold by 2,3 dimensional distribution formula [ for avoiding groups of stones in ACT2 space public traveling ... ];
_ WHICH2,
WHEN SYN, NOT
triangulate;
_ WHICH3,
WHEN SYN AND
triangulate;
。
ACT3 DNA growth pattern distribution: time + number = distance, WHERE time must be parallel time, and number must be based on natural time, ... ; Therefore, ɟ(x) = ... ; Also see: Plantation on the MOON;
。
Bernoulli distribution aka
BE(p): ɟ(x) = { p IFF x = 1, (1 - p) IFF (x
= 0),
,
,
,
... ; THIS generates µ within ц (0, 1); RETURN 1 IFF (µ <= p), 0 IFF ELSE;
。
Beta distribution aka
B(α, β): ɟ(x) = { (Γ(α + β)) / ((Γ(α))
(Γ(β))) ((x(α - 1)) (1 - x)β- 1) IFF (α > 0) AND
(β > 0) AND
(0 <= x <= 1),
,
,
,
,
... ; α = Integer(
(α)))
AND
β = Integer((β)));
THIS generates y1 from Ģ(α, 1)
AND y2
from Ģ(β, 1); x = (y1 / (y1
+ y2)); RETURN x;
。
Binomial distribution aka
BN(n, p): Probability mass function f with
random variable x can be calculated as ɟ(x) = { (( n! ) / ((n - x)! x!))
((px (1 - p))n-x) IFF (x = 0, 1, 2, ... , n-1, n),
,
,
,
,
... ; WHILE (n = Integer((n)))
AND
(0 < p < 1); THIS generates y1, y2, y3, ... , yn-1,
yn from BE(p); RETURN y1
+ y2 + y3 + ... + yn-1 + yn ;
。
Cauchy distribution aka
C(α, β):
ɟ(x) = { β / (π
(β2 + ((x - α)2 ))) IFF (α > 0) AND
(β > 0) AND
(-¥ < x < ¥),
,
,
,
,
... ; THIS generates µ within ц (0, 1); Probability density function's x is
assigned as x = α - (β / tan (π
µ)); RETURN x;
。
Chi-Square distribution aka X2(k):
IFF (z1, z2, z3, ... , zk within
N(0, 1)); y = i = 1Sk
zi2; k is degrees of freedom; ɟ(x) = { ((x(( k / 2) -
1) exp (- x / 2)) / (Γ (k / 2) 2(k / 2))) IFF (x >= 0),
,
,
,
,
... ; Mean and variance are k, 2k; THIS generates zi; i = 1, 2, 3,
... , k, within N(0, 1); RETURN (z12 + z22 +
z32 + ... + zk2);
。
Empirical distribution:
ɟ(x) = { 0 IFF x < a1, (((i - 1) / (n - 1)) + (x - ai)
/ ((n - 1) (ai + 1 - ai))) IFF ((ai <= x <= ai+1)
AND (1 <=
i <= n - 1)), 1 IFF an <= x,
,
,
... ; THIS generates µ within ц (0, 1); RETURN ai + (((n -1) µ - i +
1) (ai+1 - ai)); WHILE i = Integer(((n
- 1) µ + 1));
。
Erlang distribution: ... ;
。
Exponential distribution aka EXP(β):
ɟ(x) = { (1 / β) e-(x / β) IFF ((0 <= x <
¥ ) AND
(β > 0)), 0 IFF ELSE,
,
,
,
... ; THIS generates u within ц (0, 1); RETURN -(β (ln (u)));
。
F distribution: ... ;
。
Gamma distribution aka Ģ(α, β):
ɟ(x) =
{ ((xα - 1 e-(x / β) ) / βα Γ(α)) IFF (0
<= x < ¥ ) AND
(α > 0) AND
(β > 0)), 0 IFF ELSE,
,
,
,
... ; Ģ(α, β) WHICH (((α β) = NOT
constant) AND
((α β2) = NOT
constant)); Ģ(1, β) = exp (β); WHILE α = Integer((
));
THIS generates x = 0; REPEAT v within
(EXP(1));
x
= x + v; α=
α - (1 = ((1)));
UNTIL (α = 1); RETURN (β x);
。
Geometric distribution: ... ;
。
Logistic distribution: ... ;
。
Lognormal distribution aka
LOGN(µ,
s2):
WHILE (x is from N(µ, s2)
AND
y = exp (x)), probability density function f(y) = { (1 / (√(2
π
s y))) exp (- ((((ln y) - µ)2 ) / (2
s2))) IFF (0 <= y <
¥ ), 0 IFF ELSE,
,
,
,
... ; Mean and variance are exp (µ + (s2
/ 2)), ((exp(s2
)) - 1) exp (2µ + s2);
THIS generates z within N(0, 1); x=
(u + (s z)); RETURN exp(x);
。
Multi-normal distribution: ... ;
。
Negative Binomial distribution: ... ;
。
Normal distribution: ... ;
。
Poisson distribution aka
P(λ): ɟ(x) = { ((λx) e-λ)
/ x! IFF (x = 0, 1, 2, ... ), 0 IFF ELSE,
,
,
,
... ; Mean is λ (λ > 0); x = 0;
b = 1; Brunch: THIS generates u within ц (0,
1); b = b u; IF b >= e-λ, then (x
= x + 1) AND
GOTO the Brunch; Return x;
。
Student's t distribution: ... ;
。
Triangular distribution: ... ;
。
Uniform distribution: ... ;
。
Weibull distribution: ... ;
。
Also read: [Uncertain Programming; Baoding Liu; 1999];
Also see: Algorithm; Fuzzy Support Vector Machine, a Myanmar's imaginary dimensional hyperspace craft; Please notice that 6 parameters have been intentionally use ... ;
Also see: Distributive Law;
Notice that integer has been adjusted [cast: i.e. x = cast
(y)] by natural time;
FP, Fuzzy Parameters have been used in each
fuzzy set { ... ; Also see: Mutual Exclusion's
Set{...}, Duo-binary OSI
Draft's Set{...};
For ACT3 stage developers only: also read that number 3 behaves as semantic in JUN time, and then time to develop ACT3 stage developments ... ;
For ACT3 and ACT2 stage space mathematicians only: develop differentiable manifold in lie-groups mathematically; To do so, 1st to understand, star in Kanji writing character "sun at top, 2 green lines, 3 aqua lines", and then 2nd to understand 2,3 dimensional vector, and then design and engineer 2,3 dimensional distribution formula mathematically, 3rd to prove differentiable manifold in lie-groups;
For space engineers only: Calculate the shaded 2D region:
and how to
solve horizontal area at certain vertical height; Before understanding the
distribution of energy, M Theory of strings
must be understood;
For computer system analysts only: Calculate 2 parameters values of system time(s), and then reverse engineer WHICH distribution has been used WHILE synchronization occurs among servers in Intranet; i.e. //idlist,123:4567, /list comma randomly distributed time stamp in integer : randomly distributed time stamp in integer comma has been prompted between computer A and B in a grid with TTL, Time to live < 15 ms;
...
|
|
|
 |
|
