Normal Distribution Paranormal Distribution - - PowerPoint PPT Presentation

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Normal Distribution Paranormal Distribution - - PowerPoint PPT Presentation

Apr 06, 2024 198 likes •703 views

Normal Distribution Paranormal Distribution Anna Karlin Most Slides by Alex Tsun Agenda The Normal/Gaussian RV Closure properties of the Normal RV The standard normal CDF The Central Limit Theorem! The

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Normal Distribution Paranormal Distribution Anna Karlin Most Slides by Alex Tsun

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Agenda

  • The Normal/Gaussian RV
  • Closure properties of the Normal RV
  • The standard normal CDF
  • The Central Limit Theorem!
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The Normal/Gaussian RV

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The Normal PDF

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The Standard Normal CDF

a

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The Standard Normal CDF

a

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1 − Φ(a)

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Φ(−a) = 1 − Φ(a)

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Φ(−a)

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The Standard Normal CDF

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What about non-Standard Normals?

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We can Standardize any RV

probability students Definition of Expectation

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Normals stay normal! (Under scale+Shift)

probability students Definition of Expectation

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Closure of the normal (Under scale+Shift)

probability students Definition of Expectation

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X is normal with mean 3 and variance 9. What is

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Pr (2 < X < 5)

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Pr (X > 0)

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Pr (|X-3| > 6)

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X is normal with mean 3 and variance 9. What is

○

Pr (2 < X < 5)

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Pr (X > 0)

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Pr (|X-3| > 6)

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From to standard normal

N(µ, σ2)

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Summary: The Normal/Gaussian RV

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Normal random variables

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Closure of the normal (under addition)

probability students Definition of Expectation

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Closure of the normal (under addition)

probability students Definition of Expectation

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5.7 The Central Limit Theorem

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The Sample Mean

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The Sample Mean

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The Central Limit Theorem

Consider i.i.d. (independent, identically distributed) random vars X1, X2, X3, … Where Xi has μ = E[Xi] and σ2 = Var[Xi] Consider random variables and

X1 + X2 + . . . + Xn

1 n

n

X

i=1

Xi

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The Central Limit Theorem

Consider i.i.d. (independent, identically distributed) random vars X1, X2, X3, … Where Xi has μ = E[Xi] and σ2 = Var[Xi] As n → ∞, Restated: As n → ∞,

Mn = 1 n

n

X

i=1

Xi → N ✓ µ, σ2 n ◆

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CLT (Pictures)

Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

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CLT (Pictures)

Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

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CLT (Pictures)

Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

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CLT (Pictures)

Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

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CLT (Pictures)

Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

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CLT (Pictures)

Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

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CLT (Pictures)

Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

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CLT in the real world

CLT is the reason many things appear normally distributed Many quantities = sums of (roughly) independent random vars Exam scores: sums of individual problems People’s heights: sum of many genetic & environmental factors Measurements: sums of various small instrument errors

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CLT in the real world

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CLT in the real world

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CLT in the real world

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CLT in the real world

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CLT in the real world

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CLT (Example)

Definition of Expectation

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CLT (Example)

Definition of Expectation

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CLT (Example)

Definition of Expectation

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CLT (Example)

Definition of Expectation

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  • Suppose I asked you to estimate Pr (X = 20) using the

normal approximation.

  • Problem: Binomial is discrete, Normal is continuous.

The Continuity Correction (Idea)

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The Continuity Correction (Idea)

Definition of Expectation

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The Continuity Correction (Idea)

Definition of Expectation

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The Continuity Correction (Idea)

Definition of Expectation

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The Continuity Correction

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The Continuity Correction

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The Central Limit Theorem

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Normal random variables

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The Standard Normal CDF