SYSC4005-5001-guide Sheet For Final Exam-W2021 PDF

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Cheat Sheet: Statistical Models in SimulationSYSC4005/5001 – Discrete Simulation/ModelingLecturer: Ahmed Raoof Winter 2021NOTE: For the probability distributions, the PDF (or the PMF), CDF, mean, and the varianceequations are listed. You should know how and when to use these equations.1. Bernoulli D...

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Cheat Sheet: Statistical Models in Simulation SYSC4005/5001 – Discrete Simulation/Modeling Lecturer: Ahmed Raoof

Winter 2021

NOTE: For the probability distributions, the PDF (or the PMF), CDF, mean, and the variance equations are listed. You should know how and when to use these equations.

1. Bernoulli Distribution (Discrete):

➢

➢ ➢ ➢

𝑝, 𝑥𝑗 = 1, where 𝑗 = 1,2,3, . . , 𝑛 PMF = 𝑝(𝑥𝑗 ) = {1 − 𝑝 = 𝑞, 𝑥𝑗 = 0, where 𝑗 = 1,2,3, . . , 𝑛 0, otherwise 0, 𝑗 > 0 CDF = 𝐹(𝑥𝑗 ) = {𝑞, 0 ≤ 𝑗 < 1 1 𝑗≥1 The mean 𝐸 (𝑋 ) = 𝑝 The variance 𝑉 (𝑋 ) = 𝑝. 𝑞

2. Binomial Distribution (Discrete):

𝑛 ( ) 𝑝 𝑥 𝑞𝑛−𝑥 , 𝑥 = 0,1,2, … , 𝑛 ➢ PMF = 𝑝(𝑥 ) = { 𝑥 0 otherwise 𝑛 ∑𝑥 ( ) 𝑝𝑡 𝑞𝑛−𝑡 , 𝑡 ≥ 0 ➢ CDF = 𝐹(𝑥 ) = { 𝑡=0 𝑡 0 𝑡0 𝑡=0 ➢ The mean 𝐸 (𝑋 ) =

1

𝜃

➢ The variance 𝑉 (𝑋 ) =

1

𝑡!

𝑘𝜃 2

10. Normal Distribution (Continuous): ➢ PDF = 𝑝(𝑥 ) =

∞, 𝑎𝑛𝑑 𝜎 2 > 0

➢ CDF = 𝐹(𝑥 ) =

1

𝜎√2𝜋

−1 𝑥−𝜇 2 ( ) 2 𝜎

𝑒

, 𝑤ℎ𝑒𝑟𝑒 − ∞ < 𝑥 < ∞, −∞ < 𝜇 < 2

−1 𝑡−𝜇 1 𝑥 ( ) ∫−∞ 𝜎√2𝜋 𝑒 2 𝜎

➢ The mean 𝐸 (𝑋 ) = 𝜇 ➢ The variance 𝑉 (𝑋 ) = 𝜎 2

. 𝑑𝑡

Another way to find F(x): 𝐹 (𝑥 ) = Φ (

𝑥−𝜇

Pay attention to where is Page|3

𝑥−𝜇 𝜎

𝜎

) and use table A.3

: on the positive or the negative side: Φ(−𝑥) = 1 − Φ(𝑥 )

Cheat Sheet: Statistical Models in Simulation SYSC4005/5001 – Discrete Simulation/Modeling Lecturer: Ahmed Raoof

Winter 2021

11. Weibull Distribution (Continuous): ➢ PDF = 𝑝(𝑥 ) = {𝛼 (

𝛽 𝑥−𝑣 𝛽−1

)

𝛼

𝛽

𝑥−𝑣 ) −( 𝑒 𝛼

,

𝑥≥𝑣

0 otherwise Where −∞ < 𝑣 < ∞, 𝛽 > 0, and 𝛼 > 0 0, 𝑥...