FOR DATA SCIENCEtkemp/180A/180A-Lec25-After.pdf · of random variables-in the context oftheirjoint distribution. Let X, y be two (let's say discrete) random variables. #CXty) = &glk,

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!