Normal Distribution

Chris Gregg

CS109, Stanford University

Summer, 2026

Enough Servers?

• You are running a massive website. On the busiest minute you receive:
• Average of 10⁶ requests
• Variance of 10⁴ requests_.
• You are going to buy n servers
• Each server can handle 10,000 requests per min, otherwise you drop requests
• What is the smallest value of n such that P(drop) < 0.0001

Four Prototypical Trajectories

Announcements

Announcements

• Problem Set 3 is going out today!
• Our first midterm is next week. We will have review
• Please fill out the form for OAE accommodations ASAP if you have them!

Four Prototypical Trajectories

Review

Review: Probability Density Function

Recall the definition of a derivative:

A probability!

What do you get if you integrate over a

probability density function?

Review: Probability Density Function

The probability density function (PDF) of a continuous random variable represents the derivative of probability at a given point.

​

Units of probability divided by units of X. Integrate it to get probabilities!

Uniform and Exponential Distributions

https://chrispiech.github.io/probabilityForComputerScientists/en/part2/uniform/

Cumulative Distribution Function

A cumulative distribution function (CDF) is a “closed form” equation for the probability that a random variable is less than a given value

If you learn how to use a cumulative distribution function, you can avoid integrals!

Using CDF Example. X is Exp(λ = 1)

Probability density function

Cumulative density function

f(x)

x

x

P(1 < X < 2)

or

Did you know? Exponential is Memoryless!

12

X = time until the next event

What if s time has passed?

Did you know? Exponential is Memoryless!

13

X = time until the next event

What if s time has passed?

Did you know? Exponential is Memoryless!

14

X = time until the next event

What if s time has passed?

I am going to use these two properties later in class today

15

How Long Until the Next Earthquake

Based on historical data, major earthquakes (magnitude 8.0+) happen at a rate of 0.002 per year*. What is the probability of

a major earthquake in the next 30 years?

Y = Years until the next earthquake of magnitude 8.0+

*In California, according to the USGS, 2015

Piech, CS109, Stanford University

How Long Until the Next Earthquake: another way!

Based on historical data, major earthquakes (magnitude 8.0+) happen at a rate of 0.002 per year*. What is the probability of

a major earthquake in the next 30 years?

Y = # of Earthquakes in 30 years

*In California, according to the USGS, 2015

Piech, CS109, Stanford University

Four Prototypical Trajectories

/Review

Four Prototypical Trajectories

Big Day

NormCore: A Few Normal Examples

20

Normal Random Variable

21

Variance

Expectation

PDF

mean

variance

Carl Friedrich Gauss

• Carl Friedrich Gauss (1777-1855) was a remarkably influential�German mathematician.���������
• �Did not invent Normal distribution but rather popularized it

​

22

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

23

These are log-normal

Only if they are equally weighted and independent

Most noise is assumed normal

That is actually true…

“The simplest explanation is usually the best one”

Complexity is Tempting

25

value

probability

Probably over fitting…

* That describes the training data, but will it generalize?

Fewest Assumptions

26

value

probability

μ

σ²

* A Gaussian maximizes entropy for a given mean and variance

Simple. Will generalize

Fewest Assumptions

27

value

probability

μ

σ²

* A Gaussian makes the fewest assumptions after matching mean and variance.

Simple. Will generalize

Four Prototypical Trajectories

Normal is Beautiful!

Normal Probability Density Function

29

x

f(x)

Anatomy of a Beautiful Equation

30

probability density at x

the distance to the mean

a constant

“exponential”

sigma shows up twice

Does it look less scary like this?

31

What if you had to take the log of this function?

This means "e to the power of" and is common function in code math libraries

This means "proportional to". There is a constant but there are many cases where we don’t care what it is!

Four Prototypical Trajectories

Lets go!

Let’s Try It Out: Submarine Manufacturing

33

What fraction of the panels you manufacture will meet standards?

Your team is tasked with producing the side panels for Deep Sea Submarines. Physics requires all panels to be built to within 10 microns of 500. You check how precise your manufacturing is, and find these stats:

• Average panel thickness: 𝜇=500 microns
• Variance of the thickness: 𝜎²=36 microns

34

What fraction of the panels you manufacture will meet standards?

Let’s Try It Out: Submarine Manufacturing

Your team is tasked with producing the side panels for Deep Sea Submarines. Physics requires all panels to be built to within 10 microns of 500. You check how precise your manufacturing is, and find these stats:

• Average panel thickness: 𝜇=500 microns
• Variance of the thickness: 𝜎²=36 microns

Let’s Try It Out: Submarine Manufacturing

35

What fraction of the panels you manufacture will meet standards?

Your team is tasked with producing the side panels for Deep Sea Submarines. Physics requires all panels to be built to within 10 microns of 500. You check how precise your manufacturing is, and find these stats:

• Average panel thickness: 𝜇=500 microns
• Variance of the thickness: 𝜎²=36 microns

36

What fraction of the panels you manufacture will meet standards?

Let’s Try It Out: Submarine Manufacturing

Your team is tasked with producing the side panels for Deep Sea Submarines. Physics requires all panels to be built to within 10 microns of 500. You check how precise your manufacturing is, and find these stats:

• Average panel thickness: 𝜇=500 microns
• Variance of the thickness: 𝜎²=36 microns

Submarine Manufacturing

37

37

• Loving, not scary
• …except this time

Four Prototypical Trajectories

No closed form for the integral

39

Four Prototypical Trajectories

No closed form for F(x)

Spoiler: Numerically Solved CDF

40

A function that has been solved for numerically

* We are going to spend the next few slides getting here

The cumulative density function of any normal

Linear Transform of a Normal is… Normal!

41

Linear Transform of a Normal is… Normal!

Derivation:

42

Aside: Celsius to Fahrenheit

Average temp in Palo Alto (for some date)

in Celsius:

What is the distribution in Fahrenheit?

Let

be the temperature in Fahrenheit.

Because this is a linear transform…

10

60

20

30

40

50

Probability Density

Distribution in Celsius

Distribution in Fahrenheit

Linear Transform of a Normal is… Normal!

The cutest linear transform

This is called the “standard normal”

There is a special case of linear transform for any X:

Standard deviation, not variance!

The Standard Normal

*This is the probability density function for the standard normal

Mean (μ) = 0

Variance (σ²) = 1

Phi

*This is the probability density function for the standard normal

Φ(1.31) = 0.7054

Using Table of ɸ

Symmetry of Phi

a

-a

μ = 0

*This is the probability density function for the standard normal

Symmetry of Phi

a

-a

μ = 0

*This is the probability density function for the standard normal

Interval of Phi

d

c

ɸ(c)

ɸ(d

μ = 0

Use Z to compute F(x)

Compute F(x) via Transform

For normal distribution, F(x) is computed using the phi transform.

And here we are

Table of Φ(z) values in textbook, p. 201 and handout

CDF of Standard Normal: A function that has been solved for numerically

The cumulative density function (CDF) of any normal

Using the Phi Table

Φ(0.54) = 0.7054

Do We Have To Use The Table??

Four Prototypical Trajectories

Table is kinda old school

We Are Computer Scientists!

from scipy import stats

​

stats.norm.cdf(x, mean, std)

Every modern programming language has phi stored in a library:

Piech & Cain, CS109, Stanford University

We Are Computer Scientists!

from scipy import stats

​

stats.norm.cdf(x, mean, std)

Every modern programming language has phi stored in a library:

not variance!!!

Piech & Cain, CS109, Stanford University

I Made One For You

Practice: Submarine Manufacturing

What fraction of the panels you manufacture will meet standards?

= ?

Practice: Submarine Manufacturing

What fraction of the panels you manufacture will meet standards?

Now using the CDF!

subtract mean, divide by std. dev.

Practice: Submarine Manufacturing

What fraction of the panels you manufacture will meet standards?

Now using the CDF!

Practice: Submarine Manufacturing

Now using the CDF!

What fraction of the panels you manufacture will meet standards?

Get your Gaussian On

Piech & Cain, CS109, Stanford University

Four Prototypical Trajectories

Are you ready for something different?

• A new website design is tested out on 1M users.
• Let X be the number of users whose time on the site increases with the new design.
• The CEO will endorse the new design if X ≥ 501k.

What is P(CEO endorses change| it has no effect)?

​

Midterm: Website Testing

• A new website design is tested out on 1M users.
• Let X be the number of users whose time on the site increases with the new design.
• The CEO will endorse the new design if X ≥ 501k.

What is P(CEO endorses change| it has no effect)?

​

Midterm: Website Testing

• A new website design is tested out on 1M users.
• Let X be the number of users whose time on the site increases with the new design.
• The CEO will endorse the new design if X ≥ 501k.

What is P(CEO endorses change| it has no effect)?

​

Midterm: Website Testing

>>> math.comb(1000000,501000)

• A new website design is tested out on 1M users.
• Let X be the number of users whose time on the site increases with the new design.
• The CEO will endorse the new design if X ≥ 501k.

What is P(CEO endorses change| it has no effect)?

​

Midterm: Website Testing

>>> math.comb(1000000,501000)

ValueError: Exceeds the limit (4300 digits) for integer string conversion; use sys.set_int_max_str_digits() to increase the limit

>>> 

71

Don’t worry, Normal approximates Binomial

• Galton Board

​

​

​

• (We’ll explain why�in 2 weeks’ time)

72

Poisson Approximates Binomial, With Extreme n and p

P(X = k)

k

Piech & Cain, CS109, Stanford University

• A new website design is tested out on 1M users.
• Let X be the number of users whose time on the site increases with the new design.
• The CEO will endorse the new design if X ≥ 501k.

What is P(CEO endorses change| it has no effect)?

​

Midterm: Website Testing

🤨

Correct answer is 0.02270

Normal Approximation (with continuity correction)

75

64 65 66

65

✅

Binomial

Normal

You must perform a continuity correction when approximating a Binomial RV with a Normal RV.

Continuity correction

76

… 5 6 7 …

🤔

Continuity correction

77

… 5 6 7 …

• A new website design is tested out on 1M users.
• Let X be the number of users whose time on the site increases with the new design.
• The CEO will endorse the new design if X ≥ 501k.

What is P(CEO endorses change| it has no effect)?

​

Midterm: Website Testing

YOU ARE AMAZING!

Two Ways To Approximate The Binomial

Poisson approximation for big n, small p.

Normal approximation for big n, medium p.

Piech & Cain, CS109, Stanford University

n > 20

p < 0.05

n > 20

Var(X) > 10

Four Prototypical Trajectories

Just Invented the Normal Approximation

Stanford Admissions (a while back)

81

Strategy:

🤔

(by yourself)

1. Just Binomial
2. Poisson
3. Normal
4. None/other

Stanford Admissions

82

Strategy:

1. Just Binomial
2. Poisson
3. Normal
4. None/other

Define an approximation

Solve

Continuity�correction

⚠️

✅️

SciPy can do this

How many students should Stanford admit?

83

Admit rate: 4.3%

Yield rate: 81.9%

Why Be Normal? 68% rule

• You may have heard the statement:
• “68% of the class will fall within 1 standard deviation of the exam average.”
• In general, this is only true of normal distributions:

84

Why Be Normal? 68% rule

• You may have heard the statement:
• “68% of the class will fall within 1 standard deviation of the exam average.”
• In general, this is only true of normal distributions:

85

Four Prototypical Trajectories

Challenge

Enough Servers?

• You receive R ~ N(10^6, 10^4) requests in the busiest min
• You are going to buy n servers
• Each server can handle 10,000 requests per min, otherwise you drop requests
• What is the smallest value of n such that P(drop) < 0.0001

88

Four Prototypical Trajectories

Extra Content

Four Prototypical Trajectories

How does python sample from a Gaussian?

91

from random import *

​

for i in range(10):

mean = 5

std = 1

sample = gauss(mean, std)

print sample

3.79317794179

5.19104589315

4.209360629

5.39633891584

7.10044176511

6.72655475942

5.51485158841

4.94570606131

6.14724644482

4.73774184354

How does this work?

How Does a Computer Sample a Normal?

5

0

-5

1

CDF of the Standard Normal

How Does a Computer Sample a Normal?

Further reading: Box–Muller transform

Inverse Transform Sampling

5

0

-5

1

CDF of the Standard Normal

Step 1: pick a uniform number y

between 0,1

Step 2: Find the x such that

Sample 1:

1.201234

​

Sample 2:

-0.45123

Four Prototypical Trajectories

Relative values of a PDF

Relative Probability of Continuous Variables

How much more likely are you to complete in 10 hours than in 5?

X = time to finish pset 3

X ~ N(μ = 10, σ² = 2)

Four Prototypical Trajectories

Gaussian and ELO

Gaussian Sampling and ELO ratings

97

What is the probability that the Warriors win?

How do you model zero-sum games?

Gaussian Sampling and ELO ratings

98

Arpad Elo

Gaussian Sampling and ELO ratings

99

from scipy import stats

WARRIORS_ELO = 1657

OPPONENT_ELO = 1470

STDEV = 200

NTRIALS = 10000

​

nSuccess = 0

for i in range(NTRIALS):

w = stats.norm.rvs(WARRIORS_ELO, STDEV)

o = stats.norm.rvs(OPPONENT_ELO, STDEV)

if w > o:

nSuccess += 1

print("Warriors sampled win fraction: ", float(nSuccess) / NTRIALS)

Is there a better way?

100