MATH180A: Introduction to Probability

www.math.ucsd.edu/~ynemish/180a

This week:

homework #4 (due Friday, Nov 1, 11:59pm)

Midterm 1 regrades Wednesday, Oct 30, 8am-11pm

Today: ASV 4.1-4.2

Next: ASV 4.3

toc

CltforthebinomialtfXi - Ber ( p ) ,then µ

-

- E (Xi ) = ,E- Var (Xi ) =

and X , t - -- t Xn ~

I¥KLTforbinomial).Lets~B(mp),aFor the average

à : = ¥

CLTforbinomiali.name

M = 30 n = 60

p = Yz p = Yz

h = 100 n = 1000

p = Ya pa Yz

Normalapproximation.3.si#eWe use the approximation if hpltp) > 10

Plat sû÷ ± b)#(b) -Fla)

In particulier ,in this case

PCISN- npl stp) ) = Pll ) - 91-17=294) - t Zo.gg

PASn-nptczvnpT-pTJ-PG-lfzzo.sspllsn-nplcztnpT.pl) = 913) -Pls) > 0-99

CLT.tn/amplesFlippinga fair coin 10000 times.

A- member of Tails.

Find P( 4950 EXE 5050)

[VExam 0

Continuity correctionP ( ke Sn a- e)=P (K- tzasnelttz)Exorcise

Approximale the probability that out of 300 die rolls

we get exact tg 100 members that are multiples of 3 .

CLT.tt/amplesExercise4.21 . You win $10 with probabilités ,

you lose $1 with probability É .

Approximer probability that you lost - 100$.

after200 games .

X = member of vins .

CLT.Proofforbinomialfact.tlStirling's formula) As na- , n ! nnnêvzttn

Facts For lxkl , log ( It x) = × - E +01×3 ) § 9=1-P

Proofo.

Plat §nnqP_ e- b)= Pfavnpqtxnpesnebtnpqxnp)= ZPCSN -- K)

atnpqtnpekenptbvnpq a

Msn -- K) =

#H÷ËJake log :

Jake exp :

LawoflargenumbersCLLLe.tk, __ ,Xn

. ..

be independent and identicallydistributed and (et ECX ,)=µe R . Then

het Xin Berlp) .

TY LLN for binomial ).

Sn - BCn.pl . Then for any

Binomial today , general later

LLN.proofforbinomi.at/7¥