f A ca mm b The probability that defined is Isa as E Prew - PDF document

Random variables July 22g 2020 1 Random variables 2 Probability Distribution 3 Multiple Random Variables 4 Expectation Questions about outcomes roll two dice Experiment fly Sample g 6.6 space Clg Dsg Ig 2 g 63 What's the summation flip 100 Coins Experiment space f H H Sample Hg THA g T TT 19 Hg How many heads in 100 Coins a random student CS 70 in Experiment choose f S g Sz g Sample space g S n What his her midterm is score E hand back assignments to 3 students at Experiment random Sample space 1.23g 132g 213,231g 3123321 How many students get back their own assignment a number is In scenario n g the each outcome number is known function of The a Outcomes

1 Random Variables X For A Random Variable en Periment with an R function sample r X c space is a outcome a real number Newt to Thus N assigns r each wer wa X G wi i P d 3 d The function XL is defined on the r outcomes a variable The function not random is n not what varies at random from experiment to experiment The outcome Definitions one defines For a EIR a wer I Xcw ah f A ca mm b The probability that defined is Isa as E Prew prey Prata a WE Naka

two rolled dice The summation of number on Died piles ful Xing 81 6 Is F of what is thelikelihood 5 so 4 o oo o ofgetting a sum of g y j.gg pier I l f too whew 3 ftp.D.DE R 4 8 3 5 7 10 It 12 2 6 9 prEx 3JsPV FL3D Egg 5 83sprcx teoD prca gg 2 Probability Distribution The probability of X taking on a value a random variable X The distribution of Definition is tea Pret ay a e Af where Ac is the range of X to EtaBdBM it b oaths v SIR a b

Examples Handing back assignments at random 3 Students Hand back assignments to 8D 31 1 a GEE 13321323333133201 r i w 6 Mwc cuz How many students get back their assignments own 0 if f 3g Ig Ig 0 0 Random Variables X Cw iq TPrs3zsIz I 3 3 1 57 85 3 sorry 7 d l o a J O l 23 two rolled dice The summation of number on Prem Diez 2345654321 1 31 6I 6 ooo 5 o 3ieis sk i g I s Traps 12 a a or a o 4 3 5 8 12 6 7 10 It 2 9

Named Distributions and over again Some distributions come up over Bernoulli Distribution or with heads probability p flip a biased coin as or tails cos heads Random variable X takes l a random variably thatakes 90,4 X.sn pr Eff X f of Est P g LpifIs A Bernoulli random variable X is writen as X N Bernoulli P Binomial Distribution Flip n biased with heads probability P coins number of heads X Random variable Is0 T for Pr HE X I How many sample points in events of I heads n coin flips out heads wp.hr toif.ntheeadPgrobpabiditTofwifwha t P of tails Pr

i pgn p P sum of Pr w e Hsi Probability of Hsi is WT pref't D Prew E Prati Cassi w X i Example Binomial Theorem channels Error A Packet is corrupted with probability P Send Packets me 2k at most k Probability of corruptions X number of corruptions prcxsk3 i3 EE Pru i F piei pg.dk D j 9

3 Multiple Random variables multiple random variables may be interested in One The concept of can then be extended distribution a of for multiple for combination the values variables Definition Joint distribution discrete random The joint distribution two for variables X and Y is gbEIBfT fccasbjgprfxagybD.ae Possible values taken of all where A is the set of all values fake by X and 1B is the set by y of Pr An B A a X it where as Think and B Ysb Marginal distributio Pr is As a Then what is determined by summing Phx saz Y values all for over Keagy slot Pr P8 Ea Ee B

random variables with the more In case the joint distribution is y Xn Xi Xz then gXnsanJgaiEAig Pr Xpsa nandEEprcxraig.rgxn an I i AjEIAj j How to find Prc af prcxi geq.PrExr an oXn an3 ai3sg.EE a Example I 2 3 yX 4 5 67 PREM O 0.10 050 3 O P O O 0.15 A 0.1 al L 2 O 0.050.05 O O 0 5 o a ooo og g o o co 0 0.350.5 8 Big O A 0 O 050.050.050.10 4 f Pr I 30 050 Definition Independence Random variables x and Y independent if a andy b the events are X are independent for all values bi Pr XsaJpr Ysb3 PrExsagYsb tf azb

Example Indicators very important offipacoiimes Define Ig the indicator the for is j th flip coin are mutually independent In Is independent and is This known as dentically distributed ab ii d set of random set of i i d I indicator In Hence is a tandem variables We extensively use later indicators combining random variables Difinition X Y R V and Z be Let on r a function G 1123 IR 9 X is the R V Then y z that assigns Kew Y value 2 to the g W w w then Vcu sgCxew3Ycw3 V 9CXgYgz If 2 Cw Exampels XY 3 hey LX Y z

distributions conditional probability for tD e Pry yEalb3 Prc x al y b Preys by

4 Expectation to calculate the it is very hard sometimes complete distribution of r V a would like to summarize the distribution into convenient form that is also a more compact to easier compute The most widely used such Form is the of the RV or or average expectation mean The expected value of a random Definition Vari ble X is aaPr X sa E x s Etty Theorem X cud a PRE WT E guy WER a PrEx E Ex proof e s prew Ea a Fgxcussa EE.xcwaj.arcw3 EE.xewffawprcw3 wfzxewst.ro

Example e Roll afairdie the die X value on e f Ig 213,4 bg If r If Prew s thze Zaza Faa Prata g E Ex a s Iz 1 6 6 in the range of X doesn't have to Note EEK be The expected value is not the value that you expect Rolled Example dice two dice Summation of the rolled X a Ea PrExray 2345 6 5 43 21 31 C 6 a E errand BgBg3zs gg 2 3 4 5 67 8 9 10 7 J an I convenient Not

lDistribUtion Expectation of B.in Bin Xv hop i hi pic pi Prust s n i 7 pice p s zaPrC j C Ex not easy Linearity of Expectation For any two random variables Theorem X and y have we s EE De EET C Chey Also for any constant c CE Ecc x s In general ECC Xie CEExise enfexig eCnXn G w n

WE Zeus Prew Cnxn Proof C ax vs Xm Z E CC X cute Prew ten Xn w w EX Cw Prew Cn Exncw Prcw SC t e EE XD CnECXn C assumption is V.V There on Xp no X not need do to be r V gXn X a independent Example Rolledtwodice.X.se dice Summation of the rolled a write we can X e t2 X ng X Number on die 1 Xuenumber on die 2 EEX.tk EEXDtECXD Ect 7 ae Iz 7 linearity

Expectation of Binomial Distribution the t.si aseffTPenamPde Consider for Ith flip an indicator r V Ii is Defines being heads PrEIisDsp ith flip Beheads r s f z si P D ith flip is pros fail E X of n flips heads in Define n E Ij In I Iz X s e e Est we had I n C ExTs job tf Pie P not easy E P j up EE Io EE E EXT Ii linearity Eois iii Preity s EEX3s