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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¥