Joint Probability Mass Function
E[XY] |
Summe von x * y * f(xy) |
Arena Variables and Function
DISC(0.3, 1, 0.8, 2, 1.0, 3) |
DISC generates random discrete values based on cumulative probabilities; pair each probability with its corresponding value. |
TNOW |
Current simulated time |
NR(Res) |
Res Servers currently in service |
NQ(Queue) |
Number of customers in Queue |
Mod.NumberOut |
Customers who have left the mode |
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Expected Value, Variance, ...
Discrete E[X] |
SUM[x * f(x)] |
Continous E[X] |
SUM[x * f(x) dx] |
Variance of X |
E[X2] - (E[X])2 |
Standard Deviation of X |
SQRT[Var(X)] |
Random Number Generators
Bad generators |
Midsquare number generator, Random number tables, von Neumann's mid-square method, Fibonacci generator, Additive congruential generator, RANDU |
Good generators |
Linear Congruential Generators (modern cycle length > 2191; Mersenne Twister (219937) |
Randu |
65539Xi mod(231) |
Desert island |
16807Xi mod(231-1) mod(2147483647) |
Inverse Transform Method Key Problems
If X ~ Normal(0,1), what’s the distribution of Φ(X)? |
Unif(0,1) |
If U∼Uniform(0,1) and Φ(x) is the CDF of the standard normal, what is the distribution of 2Φ −1 (U)+3? |
Φ−1(U) turns a Uniform(0,1) into a Normal(0,1). Multiply by 2 → scales the standard deviation by 2 = Normal(0,4). Add 3 → shifts the mean to 3. = Normal(3,4) |
-3ℓn(U²V²) where U, V ~ i.i.d. Unif(0,1) |
= -6ℓn(U) - 6ℓn(V) ~ Exp(1/6) + Exp(1/6) ~ Erlang₂(1/6) |
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Given CDF with two cases, generate X
Complete Distribution Reference Table
Box Muller Method
Z1 |
√(-2·ln(U₁)) · cos(2π·U₂) |
Z2 |
√(-2·ln(U₁)) · sin(2π·U₂) |
Radian-Modus einschalten!
Chi-Square Distribution
If Z₁, Z₂, Z₃ are i.i.d. Nor(0,1), find c such that P(Z₁² + Z₂² + Z₃² < c) = 0.99
Calculator: chiSqInv(0.99, 3) → 11.34 |
Find distribution of U1 and U2
Find distribution of -4(U₁ + U₂) - 2 |
U₁ + U₂ ~ Tria(0, 1, 2) |
Apply the transformation -4(U₁ + U₂) = 4(Tria(0, 1, 2) ) |
The minimum becomes: -4(2) = -8; The mode becomes: -4(1) = -4; The maximum becomes: -4(0) = 0 |
4·Tria(0, 1, 2) = Tria(-8, -4, 0) |
Subtract 2 |
Tria(-10, -6, -2) |
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