Distribution     Last updated on 2012/2555     2     7, 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. 25 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 "stars" in ACT2 space public traveling ... ];

^_ WHICH2, WHEN SYN, NOT triangulate;

^_ WHICH3, WHEN SYN AND triangulate;

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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;

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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;

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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;

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Binomial distribution aka BN(n, p): Probability mass function f with random variable x can be calculated as ɟ(x) = { (( n! ) / ((n - x)! x!))  ((p^x (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, ... , y_n-1, y_n from BE(p); RETURN y₁ + y₂ + y₃ + ... + y_n-1 + y_n ;

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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;

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Chi-Square distribution aka X²(k): IFF (z₁, z₂, z₃, ... , z_k within N(0, 1)); y = _{i = 1}S^k z_i²; 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 z_i; i = 1, 2, 3, ... , k, within N(0, 1); RETURN (z₁² + z₂² + z₃² + ... + z_k²);

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Empirical distribution: ɟ(x) = { 0 IFF x < a₁, (((i - 1) / (n - 1)) + (x - a_i) / ((n - 1) (a_{i + 1} - a_i))) IFF ((a_i <= x <= a_i+1) AND (1 <= i <= n - 1)), 1 IFF a_n <= x, , , ... ; THIS generates µ within ц (0, 1); RETURN a_i + (((n -1) µ - i + 1) (a_i+1 - a_i)); WHILE i = Integer(((n - 1) µ + 1));

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Erlang distribution: ... ;

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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)));

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F distribution: ... ;

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Gamma distribution aka Ģ(α, β): ɟ(x) = { ((x^{α - 1} e^{-(x / β)} ) / β^α 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);

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Geometric distribution: ... ;

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Logistic distribution: ... ;

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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);

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(math) logistic distribution ... ;

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Multi-normal distribution: ... ;

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Negative Binomial distribution: ... ;

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Normal distribution: ... ;

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PDF, Probability Distribution Function: IFF Time . Space (Distance ab) PDF = (1/(b-a)) ... , WHERE Time is on X dimension, and PDF is on Y dimension;

i.e.

PDF=(1/(b-a)) ... ; Also see: 2011 August, Pg. 57, Computer, IEEE, www.computer.org;

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Poisson distribution aka P(λ): ɟ(x) = { ((λ^x) 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;

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Student's t distribution: ... ;

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Triangular distribution: ... ;

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Uniform distribution: ... ;

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Weibull distribution: ... ;

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z-distribution standard normal distribution;

standard	normal	distribution		z	-	distribution	;

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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;

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