Chapter 7 Random Variables 7.1: Discrete and Continuous Random Variables.
FOR DATA SCIENCEtkemp/180A/180A-Lec25-After.pdf · of random variables-in the context oftheirjoint...
Transcript of FOR DATA SCIENCEtkemp/180A/180A-Lec25-After.pdf · of random variables-in the context oftheirjoint...
MATH 180A : INTRO TO PROBABILITY( FOR DATA SCIENCE)
www.math.ucsd.edu/ntkemp/18oA
Today : § 8. l - 8.4
Next : § 9. l - 9.2
Homework 7 Due Wednesday,Nev 27
Exam l : graded & released . I mean 6340,St.
Dev.
20% )↳ Topics to focus on reviewing :
• Chebyshev 's Inequality• Transforming densities. NGF les p .
when it determines the distribution)
We can new come back to questions about sums µof random variables - in the context of their jointdistribution .
Let X,y be two ( let's say discrete ) random variables
.
# CX ty ) = & glk, e) pxy, 141) ← pxylk.lk Path,Ht)w
9%0= & text) pyo, Ml) = ? kpx,d.Best Px,Hk,t)
= fkfpxgck.es +El EpyxHD--
Pxlk) Pyle)
= f kpxih) + ftp.dl) e TINTIN.
theorem : For any random variables Xi,Xs,- . .,Xn,
FIX,t - - - t XD = END HIND t - - - t #Hn) .
Eg.
Sr Binh, p) .
Pls - k) = (f) pm-pfm asks n .This means S -- X
, txt - -- txn where X
,.. ,Xnn Berlp)
i.IE/S)--tElXDtlElXz)t---tlElXDENjj=lPlXj--1) =p .=
p t p t-
- - t p in of them)
= np .
A binomial is a sum of Bernoulli's (indicator r . v. ' s ) .
Lots of problems can be solved when we can express desiredevents in terms of sums of indicators .
Eg. Suppose we put 200 balls randomly into too boxes .
Whatis the expected number of empty boxes ?
Xr :-- B { Box i is empty} X : - #of empty boxes = Xi
is is too i. IEN ) = if END → too co.eduE. (Xi) = PIX ,-=D - Pl Bet is empty) = G.UK
"
÷ 13.4
Eg .
Your favorite cereal I chocolate frosted sugar bombs) comes witha Pokemon figurine . There are n 210 to collect . What isthe expected number of boxes you need to buy to collect them all ?
X = # of taxes you need to collect them all Y- Geom Ip)
X, s" " 1st one = I EH) = "p
ko Bef boxes after the X ,th needed to collect the 2nd
.
n GeomChin)'
,
in th
n
Xjz " Xj-i"
j th.
~ Geom ( ninth)~ 0.577
X - X, th t - - - TXn
E. M .
Elk) = EH,) t #LK) t - - t #kn) = I t F, tft . . - + Y
const .
I= n f l tf tf t - - - t ht) I nlnn tfn- told
In -- 2e) : LEN) -- 71 . esHn
sumsol-Variancevarlxi.TT= # lcxtxf)- (ENTITY 8¥= Bf * txt ) - flfdxjt#HIT= ( x2t2XYtY2) - ( EHft2BlxlEKtEHf)= ELK) +ZEND t #H2) - EAT- NEH# HI - EH
'
= Elk) - EAT t END - #HT t2(END - EHDEN)-
Varix ) Vari-
Def : Cov CX,Y) = Efx #half #Hlf f #W
-EMEK)
calculation)
Note ! Cov IX,X) = Var CX)
theorem : Var (X.tl/zt---tXn)=VarlXDtVarlXz)t---tVarlXJ=.qFGvCXi,X;) + E Cov Hi, XD
Hj
covariancedtndependen-Ifxi.lk,-
→ Xn are independent, then for its'
Cov ( Xi,Xj) = BIX ,-Xj)- Emi) #Nj)Exit #agio
- o.
Corday : If Xi ,Xz, . . . ,Xn are independentVarlxitkt-i.tl/n)=VarlX,)tVar1XDt---tValXD
AEg .
Sp Binh , p) [email protected].
Xjr Berlp)varlsnsenp" -P " fatwww.H.xkrlxd.qgcxjrecxit#HiT=pltptpltpt--tPitp-
- END -EAT= hpa
-
p) . =p- p'
e. Pll- p)
Independentv.s.tl#rreatedWe've seen that independent ru 's are uncorrelated .
The converse de es net held.
Eg .
X - Unif I -1,913 lie .PH#lHPlX=d= 's ) )Ya X?
EH) ' II - t ) t} ttgll )Goal,4) = FIXX) - EMEK) = o .
= EW) - Efx) #H2) x3=X.
"°
END a
&
= O .
Eg. Coupon Collector ( Revisited)Let Tn be the number of cereal boxes it takes to collectn distinct toys .
Tn = It W ,t Wz t -
- - t Wh- i
Wn n Geom land) are all independent .
Var l Tn) x fins
Reversientoth-Meawletxi.lk, →Xn be ii.d. random variables lie . sampling , but
not just Bernoullisay ElXj)=µ , Varlxj) = 62 . trials)The sampteneaw In -_ tnlxitxzt -
- - +XD.
# (xD . I X, ) t- - -+END) = M ixdep .
Valla x;) -
- troopVarun-
- t.E.vnHide!