[PDF] 10: The Normal (Gaussian) Distribution



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10: The Normal (Gaussian) Distribution

Lisa Yan, CS109, 2020 Carl Friedrich Gauss Carl Friedrich Gauss (1777-1855) was a remarkably influential German mathematician Did not invent Normal distribution but rather popularized it



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10: The Normal

(Gaussian) Distribution

Lisa Yan

April 27, 2020

1

Quick slide reference

2

Normal RV

3

10a_normal

Lisa Yan, CS109, 2020

Todayǯs the Big Day

Lisa Yan, CS109, 2020

defAn Normal random variable ܺ

Other names: Gaussianrandom variable

Normal Random Variable

5

Expectation

PDF mean

Lisa Yan, CS109, 2020

Carl Friedrich Gauss

Carl Friedrich Gauss (1777-1855) was a remarkably influential

German mathematician.

Did not invent Normal distribution but rather popularized it6

Lisa Yan, CS109, 2020

Why the Normal?

Thats what they

want you to believe

Lisa Yan, CS109, 2020

Why the Normal?

Common for natural phenomena:

height, weight, etc.

Most noise in the world is Normal

Often results from the sum of many

random variables

Sample means are distributed normally

8

Actually log-normal

(okay this one is true, well see this in 3 weeks)

Lisa Yan, CS109, 2020

0 0.05 0.1 0.15 0.2 0.25

044485256606490

Part of CS109 learning goals:

Translate a problem statement into a random variable In other words: model real life situations with probability distributions 9 value

How do you model student heights?

Suppose you have data from one classroom.

Fits perfectly!

But what about in

another classroom?

A Gaussian maximizes entropy for a

given mean and variance.

Part of CS109 learning goals:

Translate a problem statement into a random variable In other words: model real life situations with probability distributions 0 0.05 0.1 0.15 0.2 0.25

0444852566064900 44 48 52 56 60 64 90

10 value

How do you model student heights?

Suppose you have data from one classroom.

Same mean/var

Generalizes well

Lisa Yan, CS109, 2020

I encourage you to stay critical of how

to model real-world phenomena.

Common for natural phenomena:

height, weight, etc.

Most noise in the world is Normal

Often results from the sum of many

random variables

Sample means are distributed normally

Actually log-normal

Just an assumption

Only if equally weighted

(okay this one is true, well see this in 3 weeks)

Anatomy of a beautiful equation

12 normalizing constant exponential tail symmetric

Lisa Yan, CS109, 2020

Campus bikes

You spend some minutes, ܺ

between classes.

Variance of time spent: ߪ

Suppose ܺ

probability you spend ൒͸minutes traveling? 13 (call me if you analytically solve this)

Lisa Yan, CS109, 2020

Computing probabilities with Normal RVs

However, we can solve for probabilities numerically using a function Ȱ: 14

Cannot be

solved analytically|

CDF of

To get here, well first

need to know some properties of Normal RVs.

Normal RV:

Properties

15

Lisa Yan, CS109, 2020

1.Linear transformations of Normal RVs are also Normal RVs.

16

Lisa Yan, CS109, 2020

1. Linear transformations of Normal RVs

Linear transformations of X are also Normal.

Proof:

17

Lisa Yan, CS109, 2020

2. Symmetry of Normal RVs

18

Lisa Yan, CS109, 2020

Using symmetry of the Normal RV

19

1.ܼܲ

2.ܼܲ

3.ܼܲ

4.ܼܲ

5.ܼܲ

A.ܨ

B.ͳെܨ

C.ܨݖെܨ

Let ܼ̱ࣨͲǡͳwith CDF ܼܲ൑ݖൌܨ

Suppose we only knew numeric values

for ܨݖand ܨ

How do we compute the following probabilities?

Lisa Yan, CS109, 2020

Using symmetry of the Normal RV

A.ܨ

B.ͳെܨ

C.ܨݖെܨ

Symmetry is particularly useful when

computing probabilities of zero-mean

Normal RVs.

Normal RV:

Computing

probability 21

10c_normal_probs

Lisa Yan, CS109, 2020

22

CDF of the

Standard Normal, ܼ

Lisa Yan, CS109, 2020

The Standard Normal random variable ܼ

Other names: Unit Normal

CDF of ܼ

Standard Normal RV, ܼ

23

Note: not a new distribution; just

a special case of the Normal

Lisa Yan, CS109, 2020

Standard Normal Table only has

Lisa Yan, CS109, 2020

The first Standard Normal Table was

computed by Christian Kramp, French astronomer (17601826), in Analyse des RéfractionsAstronomiqueset

Terrestres, 1799

Used a Taylor series expansion to the

third power

Lisa Yan, CS109, 2020

Probabilities for a general Normal RV

we use Ȱ, the CDF for the Standard Normal ܼquotesdbs_dbs3.pdfusesText_6