SLIDE 1
Probability
3.1 Discrete Random Variables Basics
Anna Karlin Most slides by Alex Tsun
Anonymous questions
Agenda
- Intro to Discrete Random Variables
- Probability Mass Functions
- Cumulative Distribution function
- Expectation
Probability 3.1 Discrete Random Variables Basics Anna Karlin Most - - PowerPoint PPT Presentation
Probability 3.1 Discrete Random Variables Basics Anna Karlin Most - - PowerPoint PPT Presentation
Anonymous questions Probability 3.1 Discrete Random Variables Basics Anna Karlin Most slides by Alex Tsun Agenda Intro to Discrete Random Variables Probability Mass Functions Cumulative Distribution function Expectation
SLIDE 2
SLIDE 3
Flipping two coins
SLIDE 4
Random Variable
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SLIDE 5
20 balls numbered 1..20
- Draw a subset of 3 uniformly at random.
- Let X = maximum of the numbers on the 3 balls.
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unordered subsetsof 3 balls
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SLIDE 6
Random Variable
SLIDE 7
Identify those RVs
a
91,2 in drv
b
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c
42,3
drv
d
Cop
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Which cont
Whichhas Range
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a a
b
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SLIDE 8
Random Picture
SLIDE 9
Flipping two coins
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k
SLIDE 10
Flipping two coins
SLIDE 11
Flipping two coins
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SLIDE 12
Probability Mass Function (PMF)
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w Xlw
k3
SLIDE 13
Probability Mass Function (PMF)
i
SLIDE 14
20 balls numbered 1..20
- Draw a subset of 3 uniformly at random.
- Let X = maximum of the numbers on the 3 balls.
SLIDE 15
Probability Mass Function (PMF)
PIX 4
P Ew X
w
3
SLIDE 16
20 balls numbered 1..20
- Draw a subset of 3 uniformly at random.
- Let X = maximum of the numbers on the 3 balls.
- Pr (X = 20)
- Pr (X = 18)
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SLIDE 17
Cumulative distribution function(CDF)
The cumulative distribution function (CDF) of a random variable specifies for each possible real number , the probability that , that is
FX(x)
<latexit sha1_base64="sRHOyOvZfuCwr5D5z+nUz+WLjz4=">AB7nicbVDLSgNBEOz1GeMr6tHLYBDiJexGQY9BQTxGMA9IljA7mU2GzM4uM71iCPkILx4U8er3ePNvnCR70MSChqKqm+6uIJHCoOt+Oyura+sbm7mt/PbO7t5+4eCwYeJUM15nsYx1K6CGS6F4HQVK3ko0p1EgeTMY3kz95iPXRsTqAUcJ9yPaVyIUjKVmrfdVunpjHQLRbfszkCWiZeRImSodQtfnV7M0ogrZJIa0/bcBP0x1SiY5JN8JzU8oWxI+7xtqaIRN/54du6EnFqlR8JY21JIZurviTGNjBlFge2MKA7MojcV/PaKYZX/lioJEWu2HxRmEqCMZn+TnpCc4ZyZAlWthbCRtQTRnahPI2BG/x5WXSqJS983Ll/qJYvc7iyMExnEAJPLiEKtxBDerAYAjP8ApvTuK8O/Ox7x1xclmjuAPnM8f9lKOqg=</latexit>x
<latexit sha1_base64="hL+FaLtOT9luwfLW3Ut08xl3Pcw=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOeHjQA=</latexit>FX(x) = P(X ≤ x)
<latexit sha1_base64="c4X+es9QB862+1Tfu6CmKcTO2yw=">AB/HicbVBNS8NAEN3Ur1q/oj16WSxCeylJFfQiFAXxWMG2gTaEzXbTLt1swu5GkL9K148KOLVH+LNf+O2zUFbHw83pthZp4fMyqVZX0bhbX1jc2t4nZpZ3dv/8A8POrIKBGYtHEIuH4SBJGOWkrqhxYkFQ6DPS9c3M7/7SISkEX9QaUzcEA05DShGSkueWb71nOqkBq9gq+rAPiNwUvPMilW35oCrxM5JBeRoeZXfxDhJCRcYak7NlWrNwMCUxI9NSP5EkRniMhqSnKUchkW42P34KT7UygEkdHEF5+rviQyFUqahrztDpEZy2ZuJ/3m9RAWXbkZ5nCjC8WJRkDCoIjhLAg6oIFixVBOEBdW3QjxCAmGl8yrpEOzl1dJp1G3z+qN+/NK8zqPowiOwQmoAhtcgCa4Ay3QBhik4Bm8gjfjyXgx3o2PRWvByGfK4A+Mzx83bZKO</latexit>X ≤ x
<latexit sha1_base64="v7SPDaFpFd5Ee6fK5tkbtZs9vpE=">AB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3bpZhN2N2IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHsHM0nQj+hQ8pAzaqzU7pCeQPLUL1fcqjsHWSVeTiqQo9Evf/UGMUsjlIYJqnXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/m587JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSE137GZIalGyxKEwFMTGZ/U4GXCEzYmIJZYrbWwkbUWZsQmVbAje8surpFWrehfV2v1lpX6Tx1GEziFc/DgCupwBw1oAoMxPMrvDmJ8+K8Ox+L1oKTzxzDHzifP3objwE=</latexit>p
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SLIDE 18
Homeworks of 3 students returned randomly
- Each permutation equally likely
- X: # people who get their own homework
Prob Outcome w X(w) 1/6 1 2 3 3 1/6 1 3 2 1 1/6 2 1 3 1 1/6 2 3 1 1/6 3 1 2 1/6 3 2 1 1
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SLIDE 19
Probability
Alex Tsun Joshua Fan
SLIDE 20
Flipping two coins
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SLIDE 21
Expectation
SLIDE 22
Homeworks of 3 students returned randomly
- Each permutation equally likely
- X: # people who get their own homework
- What is E(X)?
Prob Outcome w X(w) 1/6 1 2 3 3 1/6 1 3 2 1 1/6 2 1 3 1 1/6 2 3 1 1/6 3 1 2 1/6 3 2 1 1
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SLIDE 23
Flip a biased coin until get heads (flips independent)
With probability p of coming up heads Keep flipping until the first Heads observed. Let X be the number of flips until done.
- Pr(X = 1)
- Pr(X = 2)
- Pr(X = k)
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SLIDE 24
Flip a biased coin until get heads (flips independent)
With probability p of coming up heads Keep flipping until the first Heads observed. Let X be the number of flips until done. What is E(X)?
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