SLIDE 1
Continuous random Variables
Anna Karlin Most Slides by Alex Tsun + Joshua Fan
Agenda
- Recap (PDFs and CDFs)
- The (Continuous) Uniform RV
- The Exponential RV
- Memorylessness
- The normal distribution
Continuous random Variables Anna Karlin Most Slides by Alex Tsun + - - PowerPoint PPT Presentation
Continuous random Variables Anna Karlin Most Slides by Alex Tsun + - - PowerPoint PPT Presentation
Continuous random Variables Anna Karlin Most Slides by Alex Tsun + Joshua Fan Agenda Recap (PDFs and CDFs) The (Continuous) Uniform RV The Exponential RV Memorylessness The normal distribution From Discrete to Continuous X
SLIDE 2
SLIDE 3
From Discrete to Continuous
SLIDE 4
PDF Intuition
X
cont
r v
fxH
Z
SLIDE 5
PDF Intuition
Fxx
PrfXex
- fxlw
dw
r e
fxH
Fx
prfxex
PRIX
x
PHX 4
119111111141
SLIDE 6
PDF Intuition
I
eat
to
txt
f
PRIX 4
IT
is
deff3
I
l
a
k p v
anti
SLIDE 7
Cumulative Distribution Functions (CDFs)
Itf
SLIDE 8
SLIDE 9
4.2 Zoo of Continuous RVs
SLIDE 10
The Uniform (Continuous) RV
Unifff
att
bp
I
i
un if
- il
E E
Unit Og
Diego
SLIDE 11
The Exponential PDF/CDF
probability students Definition of Expectation n
exeenbinofwtmeunB
ewntsynnfd
sdojt.name
1XIIJFnPoifzt
1
Exp
time
Iwaot
until
X
first event
SLIDE 12
The Exponential PDF/CDF
probability students Definition of Expectation
P
Suss
event
snow
taxi'xIIiE
a
Is
II
Prlyst
Prf Xlt
- at
e
yH Pr
Yet 1
PHY
t
- Fyftko
tao
SLIDE 13
The Exponential PDF/CDF
probability students Definition of Expectation
I
µ
c
- to
Iii
SLIDE 14
The Exponential RV Properties
probability students Definition of Expectation
fx
Y
II
x
dx
f
y
f
Van
EM
T
S permin
I
xhf xldx
x2aE
dx
Bga
SLIDE 15
The Exponential RV Properties
probability students Definition of Expectation
D
SLIDE 16
The Exponential RV
SLIDE 17
The Exponential RV
SLIDE 18
Random Picture
probability students Definition of Expectation
SLIDE 19
Memorylessness (Intuition)
Definition of Expectation
SLIDE 20
Memorylessness (Intuition)
Definition of Expectation
SLIDE 21
Memorylessness (Intuition)
Definition of Expectation
SLIDE 22
Memorylessness (Intuition)
f
x
NE
- x
co
SLIDE 23
Memorylessness (Intuition)
O
O
I
1
1.5 25
blue
- range
total
are
we
SLIDE 24
Memorylessness (Intuition)
SLIDE 25
Memorylessness of Exponential (Proof)
Definition of Expectation
pr X
X
Prc x
5
Pr X
stt
pr X
pts
e
SLIDE 26
Memorylessness of Exponential (Proof)
Definition of Expectation
SLIDE 27
example
- Time it takes to check someone out at a grocery store is
exponential with an expected value of 10 mins.
- Independent for different customers.
- If you are the second person in line, what is the
probability that you will have to wait between 10 and 20 mins.
i
tE
ENT'T
y
q
a
to
f5
Tn expito
t.IE
IoIEho fteiEoHte
7
qtxdx
I
e
e
2
SLIDE 28
example
- Time it takes to check someone out at a grocery store is
exponential with an expected value of 10 mins.
- Independent for different customers
- If you are the second person in line, what is the
probability that you will have to wait between 10 and 20 mins.
s
It's
Tamer
II
La La
a
SLIDE 29
END PIC
Alex Tsun Joshua Fan
SLIDE 30
4.3 The Normal/Gaussian Random Variable
A
bell
curve
SLIDE 31
Agenda
- The Normal/Gaussian RV
- Closure properties of the Normal RV
- The standard normal CDF
SLIDE 32
The Normal/Gaussian RV
SLIDE 33
The Normal PDF
10
r
j
ta
l I i
XnN 4,4
SLIDE 34
Random Picture
probability students Definition of Expectation
SLIDE 35
The Standard Normal CDF
a
<latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwTVu5ifEzqgxnAqelXqoxoWxMh9i1VNItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2oszYbEo2BG/5VXSrlW9i2qteVmp3+RxFOETuEcPLiCOtxBA1rAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q=</latexit> SLIDE 36
The Standard Normal CDF
a
<latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwTVu5ifEzqgxnAqelXqoxoWxMh9i1VNItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2oszYbEo2BG/5VXSrlW9i2qteVmp3+RxFOETuEcPLiCOtxBA1rAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q=</latexit>1 − Φ(a)
<latexit sha1_base64="XE2GIHEiozZRnB/MWU1YWeDNaL8=">AB8XicbVBNSwMxEJ31s9avqkcvwSLUg2W3CnosevFYwX5gu5Rsm1Ds8mSZIWy9F948aCIV/+N/+N2XYP2vpg4PHeDPzgpgzbVz321lZXVvf2CxsFbd3dvf2SweHLS0TRWiTSC5VJ8CaciZo0zDaSdWFEcBp+1gfJv57SeqNJPiwUxi6kd4KFjICDZWevTOe40Rq+CzYr9UdqvuDGiZeDkpQ45Gv/TVG0iSRFQYwrHWXc+NjZ9iZRjhdFrsJZrGmIzxkHYtFTi2k9nF0/RqVUGKJTKljBopv6eSHGk9SQKbGeEzUgvepn4n9dNTHjtp0zEiaGCzBeFCUdGoux9NGCKEsMnlmCimL0VkRFWmBgbUhaCt/jyMmnVqt5FtXZ/Wa7f5HEU4BhOoAIeXEd7qABTSAg4Ble4c3Rzovz7nzMW1ecfOYI/sD5/AF3ro95</latexit>Φ(−a) = 1 − Φ(a)
<latexit sha1_base64="UtAQTl5HemJVA3aXQpAYqzPnUOg=">AB/nicbVDLSsNAFJ3UV62vqLhyM1iEdtGSVE3QtGNywq2FtpQbqaTdujkwcxEKHgr7hxoYhbv8Odf+MkzUJbD1w4c869zL3HjTiTyrK+jcLK6tr6RnGztLW9s7tn7h90ZBgLQtsk5KHouiApZwFtK6Y47UaCgu9y+uBOblL/4ZEKycLgXk0j6vgwCpjHCgtDcyjfmvMKjWo4its17IHVEsDs2zVrQx4mdg5KaMcrYH51R+GJPZpoAgHKXu2FSknAaEY4XRW6seSRkAmMKI9TQPwqXSbP0ZPtXKEHuh0BUonKm/JxLwpZz6ru70QY3lopeK/3m9WHmXTsKCKFY0IPOPvJhjFeI0CzxkghLFp5oAEUzviskYBClE0tDsBdPXiadRt0+qzfuzsvN6zyOIjpGJ6iCbHSBmugWtVAbEZSgZ/SK3own48V4Nz7mrQUjnzlEf2B8/gAFuJLx</latexit>Φ(−a)
<latexit sha1_base64="UyhIi9uJIiIGu7TMPZg2VlqWys=">AB8HicbVBNSwMxEJ31s9avqkcvwSLUg2W3CnosevFYwX5Iu5Rsm1Dk+ySZIWy9Fd48aCIV3+ON/+N2XYP2vpg4PHeDPzgpgzbVz321lZXVvf2CxsFbd3dvf2SweHLR0litAmiXikOgHWlDNJm4YZTjuxolgEnLaD8W3mt5+o0iySD2YSU1/goWQhI9hY6bHXGLHKOT4r9ktlt+rOgJaJl5My5Gj0S1+9QUQSQaUhHGvd9dzY+ClWhFOp8VeomMyRgPadSiQXVfjo7eIpOrTJAYaRsSYNm6u+JFAutJyKwnQKbkV70MvE/r5uY8NpPmYwTQyWZLwoTjkyEsu/RgClKDJ9Ygoli9lZERlhYmxGWQje4svLpFWrehfV2v1luX6Tx1GAYziBCnhwBXW4gwY0gYCAZ3iFN0c5L8678zFvXHymSP4A+fzBweBjz4=</latexit> SLIDE 37
The Standard Normal CDF
SLIDE 38
Normal Distribution Paranormal Distribution