Upload
xavierlin
View
229
Download
0
Embed Size (px)
Citation preview
7/31/2019 01-Axioms of Probability
1/19
Chap 1 Axioms of Probability
Ghahramani 3rd edition
7/31/2019 01-Axioms of Probability
2/19
p2.
Outline
1.1 I nt roduct ion1.2 Sample space and events
1.3 Axioms of probabilit y
1.4 Basic Theorems
1.5 Cont inuit y of probabil it y funct ion
1.6 Probabili t ies 0 and 11.7 Random select ion of point s f rom int ervals
7/31/2019 01-Axioms of Probability
3/19
p3.
1.1 Introduction
Advent of Probabili t y as a math discipline
1. 4-sided die: Ancient Egypt ; 3500 B.C.
2. 6-sided die: 1600 B.C.
3. Dice: China; 7t h-10th centur ies
4. Playing cards: Much more recent
7/31/2019 01-Axioms of Probability
4/19
p4.
Introduction
St udies of Chances of Event sI ta ly
Luca Paccioli (1445-1514)
Niccolo Tart aglia(1499-1557)
Girolamo Cardano(1501-1576)
Galileo Galielei(1564-1642)
France
Blaise Pascal(1623-1662)Pierre de Fermat(1601-1665)
Chr ist ian Huygens(1629-1695) Dutch
1657 fi rst prob. bookOn Calculat ions in Games of Chance"
7/31/2019 01-Axioms of Probability
5/19
p5.
Introduction
James Bernoull i( 1654-1705)Abraham de Moivre(1667-1754)
Pierre-Simon Laplace(1749-1827)
Sim eon Denis Poisson( 1781-1840)Karl Fr iedr ich Gauss(1777-1855)
RussiaPafnut y Chebyshev(1821-1894)
Andrei Markov(1856-1922)
Aleksandr Lyapunov(1857-1918)
7/31/2019 01-Axioms of Probability
6/19
p6.
Introduction
1900 David Hilbert (1862-1943) pointed out t he
problem of the axiomat ic t reat ment of the t heory
of probabil i t y
Emile Borel(1871-1956)
Serge Bernst ein( 1880-1968)
Richard von Mises(1883-1953)
* 1933 Andrei Kolmogorov( 1903-1087) Russian
successfully ax iomat ized t he t heory of probabili t y
7/31/2019 01-Axioms of Probability
7/19
p7.
1.2 Sample space and events
Experiment (eg. Tossing a die)
Outcome(sample point )
Sample space= { al l outcomes}
Event : subset of sample space
Ex1.1 tossing a coin once
sample space S = { H, T}
Ex1.2 fl ipping a coin and t ossing a die if T
or f l ipping a coin again i f H
S= { T1,T2,T3,T4,T5,T6,HT,HH}
7/31/2019 01-Axioms of Probability
8/19
p8.
Sample space and events
Ex1.3 measuring t he l i fet ime of a l ight bulb
S= { x : x 0}
E= { x: x 100} is t he event t hat t he l ightbulb last s at least 100 hours
Ex1.4 all families w it h 1, 2, or 3 children(genders specif ied)
S= { b,g,bg,gb,bb,gg,bbb,bgb,bbg,bgg,
ggg,gbg,ggb,gbb}
7/31/2019 01-Axioms of Probability
9/19
p9.
Sample space and events Event E has occurred in an experiment :
I f t he out come of an experim ent belongs t o E.
Take event s E, F as sets and sample space S t hen
can be defined st raight forw ard.
Can also def ine
if { E1, E2, } is a set of event s
ESEFEFEFEEFFE c ,,,,
iiiiinii
ni EEEE
1111
,,,
7/31/2019 01-Axioms of Probability
10/19
p10.
Sample space and events
Associat ive law s EU(FUG)= (EUF)UG Dist ribut ive law s (EF)UH= (EUH)(FUH)
(EUF)H= (EH)U(FH)
De Morgans 1 st law : (E U F) c = EcFc
De Morgans 2nd law : (EF) c = Ec U Fc
E = ES = E(FUFc) = EF U EFc
7/31/2019 01-Axioms of Probability
11/19
p11.
1.3 Axioms of probabilityDefinit ion(Probabilit y Axioms)
S: sample space
A: event ,
Pr: a funct ion for each event A, i.e. Pr : 2S R
Pr(A) is said to be t he probabilit y of A if
Axiom 1 Pr (A) > = 0
Axiom 2 Pr(S) = 1
Axiom 3 I f { A1, A2, A3, } is a sequence of
mut ually exclusive events t hen
SA
)Pr()Pr(11
i iii
AA
7/31/2019 01-Axioms of Probability
12/19
7/31/2019 01-Axioms of Probability
13/19
p13.
Basic Theorem
Ex 1.15 I n a comm unit y of 400 adult s, 300 bike orsw im or do bot h, 160 sw im, and 120 sw im and bike.What is t he probabilit y t hat an adult , select ed atrandom from t his comm unit y, bike?
Sol: A: event t hat t he person sw ims
B: event t hat t he person bikes
P(AUB)= 300/ 400, P(A)= 160/ 400,
P(AB)= 120/ 400
P(B)= P(AUB)+ P(AB) -P(A)
= 300/ 400+ 120/ 400-160/ 400= 260/ 400= 0.65
7/31/2019 01-Axioms of Probability
14/19
p14.
Basic Theorem Ex 1.16 A number is chosen at random from t he
set of numbers { 1, 2, 3, , 1000} . What is t heprobabil i t y t hat it is divisible by 3 or 5( I .e. eit her3 or 5 or both)?
Sol: A: event t hat t he out come is divisible by 3
B: event t hat t he out come is divisible by 5
P(AUB)= P(A)+ P(B) -P(AB)
= 333/ 1000+ 200/ 1000-66/ 1000= 467/ 1000
7/31/2019 01-Axioms of Probability
15/19
p15.
Basic Theorem
I nclusion-Exclusion Pr inciple
)...()1(...)(
)()()...(
211
21
nn
kji
jiin
AAAPAAAP
AAPAPAAAP
7/31/2019 01-Axioms of Probability
16/19
p16.
1.5 Continuity of probabilityfunction
Recall t he cont inuit y of a funct ion f : R R
fro every convergent seq { xn} in R.
The cont inuit y of probabilit y funct ion is sim ilar.
Def. A seq { En, n> = 1} of event of a sample space
is called increasing i f
it is called decreasing if
)lim()(lim n
n
n
n
xfxf
;1321 nn EEEEE
.1321 nn EEEEE
7/31/2019 01-Axioms of Probability
17/19
p17.
Continuity of probability function Thm 1.8(cont inuit y of probabil i t y function)
For any increasing or decreasing sequence of
events, { En, n> = 1} :
l im P(En)= P( l im En)
7/31/2019 01-Axioms of Probability
18/19
p18.
1.6 Probabilities 0 and 1 I f E and F are events w it h probabili t ies 1 and 0,
t hen it is not correct t o say t hat E is t he sample
space S and F is the empt y set .
Example: select ing a random point from ( 0,1)
1. A= { 1/ 3} , P( A) = 0
2. B= (0,1)-A, P(B)= 1
7/31/2019 01-Axioms of Probability
19/19
p19.
1.7 Random selection of pointsfrom intervals Def. A point is randomly selected from an interval
(a, b) . The probabili t y t he subint erval ( c, d)
contains t he point is defined t o be (d-c) / (b-a).