AE4426-19 Stochastic Processes And Simulation at Delft University Of Technology | Flashcards & Summaries

# Lernmaterialien für AE4426-19 Stochastic Processes and Simulation an der Delft University of Technology

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probability mass function (pdf)

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Let pX(x) denote the probability mass function (pdf) of X, where pX(x) : R → [0, 1], pX(x) = P(X = x).

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Time-homogeneous Markov chains are processes where the probability of transition

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is independent of the time t of the transition, i.e.,

pXt+1|Xt (j|i) = pXt|Xt−1 (j|i), ∀t.

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What is the (long-run) fraction of time we stay in a state?

Ergodic

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Stationary probability distribution

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Note that P(a < X ≤ b) =

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FX(b) − FX(a).

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The variance possesses the following properties:

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1.Var(aX) = a2Var(X).

2.Var(aX + bY) = a2Var(X) + b2Var(Y) + 2abCov(X,Y).

When X and Y are independent r.v., then Cov(X, Y) = 0.

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If A, B ∈ F and P(B) > 0, the conditional probability of A given B is denoted and deﬁned by

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P(A|B) = P(A ∩ B) P(B) .

Consequence: P(A ∩ B) = P(A|B)P(B). When A and B are independent events: P(A ∩ B) = P(A)P(B).

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The variance Var[X] of X is deﬁned as:

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Var[X] = E[(X − E[X])2] = E[X2] − (E[X])2.

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Suppose C1, C2, . . . , Cm are mutually exclusive events with C1, C2, . . . , Cm a partition of Ω.

Theorem (Bayes’ rule)

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For any event A with P(A) > 0: P(B|A) = P(A,B)/ P(A) = P(A|B)P(B)/ P(A) .

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Monte Carlo simulation →

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repeated sampling of a random variable(s).

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Key theorems for Monte Carlo simulation:

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1 Law of Large Numbers (LLN)

2 Central Limit Theorem (CLT)

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CI

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Conﬁdence Intervals

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Suppose C1, C2, . . . , Cm are mutually exclusive events with C1, C2, . . . , Cm a partition of Ω.

Theorem (Partition theorem/Law of Total Probability)

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For any event A: P(A) = P(A|C1) + . . . + P(A|Cm).

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## Beispielhafte Karteikarten für deinen AE4426-19 Stochastic Processes and Simulation Kurs an der Delft University of Technology - von Kommilitonen auf StudySmarter erstellt!

Q:

probability mass function (pdf)

A:

Let pX(x) denote the probability mass function (pdf) of X, where pX(x) : R → [0, 1], pX(x) = P(X = x).

Q:

Time-homogeneous Markov chains are processes where the probability of transition

A:

is independent of the time t of the transition, i.e.,

pXt+1|Xt (j|i) = pXt|Xt−1 (j|i), ∀t.

Q:

What is the (long-run) fraction of time we stay in a state?

Ergodic

A:

Stationary probability distribution

Q:

Note that P(a < X ≤ b) =

A:

FX(b) − FX(a).

Q:

The variance possesses the following properties:

A:

1.Var(aX) = a2Var(X).

2.Var(aX + bY) = a2Var(X) + b2Var(Y) + 2abCov(X,Y).

When X and Y are independent r.v., then Cov(X, Y) = 0.

Q:

If A, B ∈ F and P(B) > 0, the conditional probability of A given B is denoted and deﬁned by

A:

P(A|B) = P(A ∩ B) P(B) .

Consequence: P(A ∩ B) = P(A|B)P(B). When A and B are independent events: P(A ∩ B) = P(A)P(B).

Q:

The variance Var[X] of X is deﬁned as:

A:

Var[X] = E[(X − E[X])2] = E[X2] − (E[X])2.

Q:

Suppose C1, C2, . . . , Cm are mutually exclusive events with C1, C2, . . . , Cm a partition of Ω.

Theorem (Bayes’ rule)

A:

For any event A with P(A) > 0: P(B|A) = P(A,B)/ P(A) = P(A|B)P(B)/ P(A) .

Q:

Monte Carlo simulation →

A:

repeated sampling of a random variable(s).

Q:

Key theorems for Monte Carlo simulation:

A:

1 Law of Large Numbers (LLN)

2 Central Limit Theorem (CLT)

Q:

CI

A:

Conﬁdence Intervals

Q:

Suppose C1, C2, . . . , Cm are mutually exclusive events with C1, C2, . . . , Cm a partition of Ω.

Theorem (Partition theorem/Law of Total Probability)

A:

For any event A: P(A) = P(A|C1) + . . . + P(A|Cm).

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