Probability and Statistics for Computer Science Can we call - - PowerPoint PPT Presentation

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Probability and Statistics for Computer Science

Can we call the exci-ng ?

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Midterm

1 Oct

Thurs

( CBT F )

12:30pm

conflicts

&

alternative

Oct

8

8pm

→

CBTF

Oct

9, 8 am

→

CBTF

Oct

8,

4pm

check

• ut

instructions , practices

Last time

Poisson Distribution

continuous

Random

Variable

probability Density

Function

Objectives

Cumulative

Distribution

Function

for

continuous random

variable

Normal

Exponential

Distribution

Pdt

f

poxidx =/

• o

Cumulative

Distribution

Function

Det

P ( X E x )

Discrete Rv

PCXEXK.IE?pcX=x )

• 2
• S

continuous

RV

×

pcxex ) =)

pix, dx

^

→ ¥1,1

"

• x

Cumulative distribution of continuous uniform distribution

Cumula-ve distribu-on func-on (CDF)

• f a uniform random variable X is:

X

b a 1

p(x)

1 b − a

X

b a

CDF P(X ≤ x) = x

−∞

p(x)dx

1

• m¥

,

• -T

Q: Probability density function: spinner

What is the constant c given the spin angle θ

has the following pdf? θ

2π

p(θ)

π

c

• A. 1
• B. 1/π
• C. 2/π
• D. 4/π
• E. 1/2π
• ,

what is the

CDF

?

a

÷

,

Quantile

( DF

(

• 7

K

4.on to 7

Normal (Gaussian) distribution

The most famous con-nuous random variable

distribu-on. The probability density is this:

p(x) = 1 σ √ 2π exp(−(x − µ)2 2σ2 )

[Trix) dx

• r
• b

(exp) →e

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Normal (Gaussian) distribution

The most famous con-nuous random variable

distribu-on. The probability density is this:

p(x) = 1 σ √ 2π exp(−(x − µ)2 2σ2 )

E[X] = µ & var[X] = σ2

Normal (Gaussian) distribution

A lot of data in nature are approximately

normally distributed, ie. Adult height, etc. p(x) = 1 σ √ 2π exp(−(x − µ)2 2σ2 )

E[X] = µ & var[X] = σ2

PDF and CDF of normal distribution curves

probability density tune .

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( PDF )

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Quantile)

Quan(les)give)

a)measure)of) loca(on,)the) median)is)the) 0.5)quan(le)

• ,

/

2=0.253

OFI

fo . 6=0.253

y

• ÷

Q.)

What)is)the)value)of)50%)quan(le)in)a)

standard)normal)distribu(on?)

• A. J1)
• B. 0)
• C. 1)
• "

Spread of normal (Gaussian) distributed data

99.7% 95% 68%

res

f p CX) DX

si:.mx I

Standard normal distribution

If we standardize the normal distribu-on (by

subtrac-ng μ and dividing by σ), we get a random variable that has standard normal distribu-on.

A con-nuous random variable X is standard

normal if

p(x) = 1 √ 2π exp(−x2 2 )

x -U

g-

Derivation of standard normal distribution

p(x) = 1 √ 2π exp(−x2 2 )

+∞

p(x) dx = +∞

1 σ √ 2π exp(−(x − µ)2 2σ2 ) dx = +∞

1 σ √ 2π exp(− ˆ x2 2 )σ dˆ x = +∞

1 √ 2π exp(− ˆ x2 2 ) dˆ x = +∞

p(ˆ x) dx

Call this standard and omit using a hat ˆ x = x − µ σ

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• Q. What is the mean of standard normal?
• A. 0
• B. 1

e

• Q. What is the standard deviation of

standard normal?

• A. 0
• B. 1

a

Standard normal distribution

If we standardize the normal distribu-on (by

subtrac-ng μ and dividing by σ), we get a random variable that has standard normal distribu-on.

A con-nuous random variable X is standard

normal if

p(x) = 1 √ 2π exp(−x2 2 )

E[X] = 0 & var[X] = 1

Another way to check the spread of normal distributed data

Frac-on of normal data within 1 standard

devia-on from the mean.

Frac-on of normal data within k standard

devia-ons from the mean.

1 √ 2π 1

exp(−x2 2 )dx ≃ 0.68

1 √ 2π k

exp(−x2 2 )dx 1=5

Using the standard normal’s table to calculate for a normal distribution’s probability

If X ~ N (μ=3, σ2 =16) (normal distribu-on) P(X ≤ 5) =?

→ I

ss

#

• - -

÷÷÷

Q.

If X ~ N (μ=3, σ2 =16) (normal distribu-on) P(X ≤ 5) =?

A . 0.5199 B. 0.5987 C. 0.6915

a-

• Q. Is the table with only positive x

values enough?

• A. Yes B. No.

e

Central limit theorem (CLT)

The distribu-on of the sum of N independent

iden-cal (IID) random variables tends toward a normal distribu-on as N

Even when the component random variables

are not exactly IID, the result is approximately true and very useful in prac-ce

∞

a S

OI=CDF

S

• t XN

,

µ→s

s →

• tinsel S nI→ E c Norma,

Central limit theorem (CLT)

CLT helps explain the prevalence of normal

distribu-ons in nature

A binomial random variable tends toward a

normal distribu-on when N is large due to the fact it is the sum of IID Bernoulli random variables

The Binomial distributed beads of the Galton Board

The Binomial distribu-on looks very similar to Normal when N is large

Binomial approximation with Normal

• ● ● ●
• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

• ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ● ● ● ● ● ●

Binomial distribu-on

μ = 20, σ2 = 10 n= 40, p=0.5

• ● ● ● ● ● ● ● ● ● ● ● ●
• ● ● ● ● ● ● ● ● ● ● ● ●

Approxima-on with Normal

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pix-ks-cyeypkci.pt

• k

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XB =

Bernoulli

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Binomial approximation with Normal

E[k] = np = 40 · 0.5 = 20 P(10 ≤ k ≤ 25) =

• k=10

40 k

• 0.5k0.540−k

=

• k=10

40 k

• 0.540 ≃ 0.96

std[k] =

• np(1 − p)

= √ 40 · 0.5 · 0.5 = √ 10 Let k be the number of heads appeared in 40

tosses of fair coin

The goal is to es-mate the following with normal

• -

Binomial approximation with Normal

P(10 ≤ k ≤ 25) ≃ 1 σ √ 2π 25

exp(−(x − µ)2 2σ2 )dx = 1 √ 2π

• 25−20

exp(−x2 2 )dx ≃ 0.94

Use the same mean and standard devia-on of

the original binomial distribu-on.

Then standardize the normal to do the

calcula-on

σ = √ 10 ≃ 3.16 µ = 20

JV (o, I)

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ii.

Exponential distribution

Common

Model for wai-ng -me

Associated

with the Poisson distribu-on with the same λ

p(x) = λe−λx for x ≥ 0

Exponential distribution

A con-nuous random variable X is exponen-al

if it represent the “-me” un-l next incident in a Poisson distribu-on with intensity λ. Proof See Degroot et al Pg 324.

It’s similar to Geometric distribu>on – the

discrete version of wai-ng in queue

p(x) = λe−λx for x ≥ 0

Expectations of Exponential distribution

A con-nuous random variable X is exponen-al

if it represent the “-me” un-l next incident in a Poisson distribu-on with intensity λ.

p(x) = λe−λx for x ≥ 0

E[X] = 1 λ & var[X] = 1 λ2

Example of exponential distribution

How long will it take un-l the next call to be

received by a call center? Suppose it’s a random variable T. If the number of incoming call is a Poisson distribu-on with intensity λ = 20 in an hour. What is the expected -me for T?

Q:

A store has a number of customers coming on

• Sat. that can be modeled as a Poisson

distribu-on. In order to measure the average rate of customers in the day, the staff recorded the -me between the arrival of customers, can he reach the same goal?

• A. Yes B. No

Additional References

Charles M. Grinstead and J. Laurie Snell

"Introduc-on to Probability”

Morris H. Degroot and Mark J. Schervish

"Probability and Sta-s-cs”

See you next time

See You!