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

Flipping two coins

Random Variable r X w io.ir rx

subsets of 3 balls unordered I 20 balls numbered 1..20 Pcwf ● Draw a subset of 3 uniformly at random. ● Let X = maximum of the numbers on the 3 balls. 7 X 2,977 3 Xiao 15 X 3,844 3,415 120 Isupportgxf 11 1 r 203 a 20 b HTT d F

Random Variable

Identify those RVs drv 91,2 in a b drr Loir c drv 42,3 d Cop Crv Whichhas Range Which cont 42,3 a a b b d

Random Picture

Flipping two coins D k X wlXlw K

Flipping two coins

Flipping two coins O

Probability Mass Function (PMF) k3 w Xlw P

Probability Mass Function (PMF) i

20 balls numbered 1..20 ● Draw a subset of 3 uniformly at random. ● Let X = maximum of the numbers on the 3 balls.

Probability Mass Function (PMF) PIX 4 3 P Ew X w

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) a Kaka ● Pr (X = 18) D X sof sewlmaxiiiihmsaaog.DK c Ma d sq K pcx ao k ao

k esr.ro <latexit sha1_base64="sRHOyOvZfuCwr5D5z+nUz+WLjz4=">AB7nicbVDLSgNBEOz1GeMr6tHLYBDiJexGQY9BQTxGMA9IljA7mU2GzM4uM71iCPkILx4U8er3ePNvnCR70MSChqKqm+6uIJHCoOt+Oyura+sbm7mt/PbO7t5+4eCwYeJUM15nsYx1K6CGS6F4HQVK3ko0p1EgeTMY3kz95iPXRsTqAUcJ9yPaVyIUjKVmrfdVunpjHQLRbfszkCWiZeRImSodQtfnV7M0ogrZJIa0/bcBP0x1SiY5JN8JzU8oWxI+7xtqaIRN/54du6EnFqlR8JY21JIZurviTGNjBlFge2MKA7MojcV/PaKYZX/lioJEWu2HxRmEqCMZn+TnpCc4ZyZAlWthbCRtQTRnahPI2BG/x5WXSqJS983Ll/qJYvc7iyMExnEAJPLiEKtxBDerAYAjP8ApvTuK8O/Ox7x1xclmjuAPnM8f9lKOqg=</latexit> <latexit sha1_base64="hL+FaLtOT9luwfLW3Ut08xl3Pcw=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOeHjQA=</latexit> <latexit sha1_base64="c4X+es9QB862+1Tfu6CmKcTO2yw=">AB/HicbVBNS8NAEN3Ur1q/oj16WSxCeylJFfQiFAXxWMG2gTaEzXbTLt1swu5GkL9K148KOLVH+LNf+O2zUFbHw83pthZp4fMyqVZX0bhbX1jc2t4nZpZ3dv/8A8POrIKBGYtHEIuH4SBJGOWkrqhxYkFQ6DPS9c3M7/7SISkEX9QaUzcEA05DShGSkueWb71nOqkBq9gq+rAPiNwUvPMilW35oCrxM5JBeRoeZXfxDhJCRcYak7NlWrNwMCUxI9NSP5EkRniMhqSnKUchkW42P34KT7UygEkdHEF5+rviQyFUqahrztDpEZy2ZuJ/3m9RAWXbkZ5nCjC8WJRkDCoIjhLAg6oIFixVBOEBdW3QjxCAmGl8yrpEOzl1dJp1G3z+qN+/NK8zqPowiOwQmoAhtcgCa4Ay3QBhik4Bm8gjfjyXgx3o2PRWvByGfK4A+Mzx83bZKO</latexit> <latexit sha1_base64="v7SPDaFpFd5Ee6fK5tkbtZs9vpE=">AB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3bpZhN2N2IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHsHM0nQj+hQ8pAzaqzU7pCeQPLUL1fcqjsHWSVeTiqQo9Evf/UGMUsjlIYJqnXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/m587JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSE137GZIalGyxKEwFMTGZ/U4GXCEzYmIJZYrbWwkbUWZsQmVbAje8surpFWrehfV2v1lpX6Tx1GEziFc/DgCupwBw1oAoMxPMrvDmJ8+K8Ox+L1oKTzxzDHzifP3objwE=</latexit> P x k 3 k p f otherwise 0 Cumulative distribution function(CDF) The cumulative distribution function (CDF) of a random variable specifies for each possible real number , F X ( x ) x the probability that , that is X ≤ x F X ( x ) = P ( X ≤ x ) x 9 as

Homeworks of 3 students returned randomly EE ta PxH ● Each permutation equally likely I k 3 ● X: # people who get their own homework 6 Prob Outcome w X(w) 1/6 1 2 3 a 3 D 1/6 1 3 2 1 pmf 1/6 2 1 3 D 1 se CDF 1/6 2 3 1 0 1/6 3 1 2 0 ya O 1/6 3 2 1 1 43 Ko x f L L p s prexex

Probability Alex Tsun Joshua Fan

Flipping two coins I Io Pratt 2 pcx 2 F O Prato X 2 Ly I I I O tyt expectation or expected value Put txt HTT P TT FIX Htt PLAID tX t th Mw tt t.tftttqt2 o

Expectation

Homeworks of 3 students returned randomly fo.is ● Each permutation equally likely ● X: # people who get their own homework PxXrPHE gkx ● What is E(X)? Px9 3 PCX Dt3PCX tl O.pfX o Prob Outcome w X(w) X 3 to l Iz 1/6 1 2 3 3 t 0 T t 1/6 1 3 2 1 I 1/6 2 1 3 1 1/6 2 3 1 0 1/6 3 1 2 0 1/6 3 2 1 1 ICX nXHPlw X 312 P 312 X 231 P 231 tX 321 P 321 t X 2B P 2B X 123 P123 X 132 P 132 3 Lg

6 I 6 ok Ey A woo Flip a biased coin until get heads (flips independent) 3 r It TITTY valuesyx With probability p of coming up heads Keep flipping until the first Heads observed. Let X be the number of flips until done. 1 ● Pr(X = 1) p a pk ● Pr(X = 2) 4 p p CtpY'p ftp b ● Pr(X = k) m

ws r ftp.wtp Flip a biased coin until get heads (flips independent) zto 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)? p Eik Pretty F X EEK p extra t.E.xi.IT EyEnia Fxr