Anna Karlin Most Slides by Alex Tsun Poisson RV Example The Zoo of - - PowerPoint PPT Presentation

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Anna Karlin Most Slides by Alex Tsun Poisson RV Example The Zoo of - - PowerPoint PPT Presentation

Dec 23, 2023 279 likes •867 views

Anna Karlin Most Slides by Alex Tsun Poisson RV Example The Zoo of Discrete RVs The Negative Binomial RV The Bernoulli RV The Hypergeometric RV The Binomial RV probability students The Geometric RV Definition of The

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Anna Karlin Most Slides by Alex Tsun

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Poisson RV Example

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The Zoo of Discrete RV’s

probability students Definition of Expectation

  • The Bernoulli RV
  • The Binomial RV
  • The Geometric RV
  • The Uniform RV
  • The Poisson RV
  • The Negative Binomial RV
  • The Hypergeometric RV
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random variables

Important Examples: Uniform(a,b): Bernoulli(p): P(X = 1) = p, P(X = 0) = 1-p μ = p, σ2= p(1-p) Binomial(n,p) μ = np, σ2 = np(1-p) Poisson(λ): μ = λ, σ2 = λ Bin(n,p) ≈ Poi(λ) where λ = np fixed, n →∞ (and so p=λ/n → 0) Geometric(p) P(X = k) = (1-p)k-1p μ = 1/p, σ2 = (1-p)/p2

!X

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Probability

4.1 Continuous random Variables Basics

Anna Karlin Most Slides by Alex Tsun

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Agenda

  • Probability Density Functions (PDFs)
  • Cumulative Distribution Functions (CDFs)
  • From Discrete to Continuous
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CDF Intuition

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CDF Intuition

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CDF Intuition

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PDF Intuition

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PDF Intuition

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PDF Intuition

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PDF Intuition

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PDF Intuition

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PDF Intuition

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PDF Intuition

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Probability Density Functions (PDFs)

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Random Picture

probability students Definition of Expectation

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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CDF Intuition

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Cumulative Distribution Functions (CDFs)

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From Discrete to Continuous

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Probability

4.2 Zoo of Continuous RVs

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Agenda

  • The (Continuous) Uniform RV
  • The Exponential RV
  • Memorylessness
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The (Continuous) Uniform RV

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The Uniform (Continuous) RV

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The Exponential PDF/CDF

probability students Definition of Expectation

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The Exponential PDF/CDF

probability students Definition of Expectation

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The Exponential PDF/CDF

probability students Definition of Expectation

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The Exponential RV Properties

probability students Definition of Expectation

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The Exponential RV Properties

probability students Definition of Expectation

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The Exponential RV

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The Exponential RV

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Random Picture

probability students Definition of Expectation

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Memorylessness (Intuition)

Definition of Expectation

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Memorylessness (Intuition)

Definition of Expectation

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Memorylessness (Intuition)

Definition of Expectation

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Memorylessness (Intuition)

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Memorylessness (Intuition)

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Memorylessness (Intuition)

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Memorylessness of Exponential (Proof)

Definition of Expectation

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Memorylessness of Exponential (Proof)

Definition of Expectation

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Memorylessness of Exponential (Proof)

Definition of Expectation

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The Gamma RV

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END PIC

Alex Tsun Joshua Fan