11
Random
experiment
Probability
Counting
→
↳ consider multiple
Product rule
If
:
Generalization ⇒
Permutation
•
all
N
stages
! ✗ hz ! ✗
Combination
↳
distinct
to order
.
_
.
)!
✗
group
A#
is
n
PCE / F)
=
partition
r
items
Probability
=
sample space
to:P(A) =P (AIB ) PCB) t
Bayes
Independent Events :
↳
Mutually exclusive
Formula
↳
and
M
n
,
.
total
objects where
n
n , one
a
#
=
0
{Fi
alike , Nz
.
.
.
selection does not matter
.
(1)
=
of
group
that
r
given
F occurred
PCE / F)
,
,
Fz ,
.
.
.
,
PCE I F)
PCE F)
=
Az)
PCA 1) P(Az Ai )
Fn } is
PCE / F)
,
mutually exclusive
PCFIEJPCEJ
=
,
,
=
=
→
1 if
group
of
F becomes
r
the
that does not
.
"
sample space
=
,
,
and
F
if
one
occur
=
,
.
.
.
PCE IF;) PCF;)
27--1 PCE / Fi ) PC Fi)
PCE F)
PCE ) PCF)
.
=
=
.
,
.
,
=
.
At AzAs)
≤ ¥ , PCEI Fit PEE it
=
"
FCE
PCAAZA } ) =P (A) PLAY A) PCA}
then PCE)
P (Fj / E)
I
PC F)
Event E
☒
if ECF
=
t #
×
contains
Ppɥ
=
PCA
:
:
×
object
as
PfAli:3 4. PCB 4
PCE / F)
:
that the order of
such
group
independent
cannot
together / independent events are unaffected
If Pl Ai) =p PCAa) =p
,PCAn)=pn ⇒ P (A) 1T¥ , ( pi )
k
trials
(E) pka-p.tn
e.g n in dependant trials success prob =p K successful
.
.
!
-
↳
m n
=
!
particular objet
label
:
of E-
PCEF)
0 if
of
.
+
=
argument
of
Cr
n
Sequential Conditioning formula
if
☒re
T.li?=ilAkl--1AilX1AzIx...xIAvl
:
↳
Conditional
.
(E) (4)
'
#
=
K members from
!
r
Pascal 's Identity :(F)
Combinatorial
↳
realizes
✗ Nr !
Selecting
:
r
items
the outcomes
Number of different permutations of
:
=
-
n
stage
important
one
n !
Cn
experiment
!
n
Ri
outcome before random
experiments
stages of which
has two
where outcome of Kth
stages
Multinomial coefficient
↳
specific
Joint
and
the distinct items
identifying
unknown before realization
i
possible outcomes
experiment
ways
:
outcomes
Quantifies chance of
→
enumerating
-
multiple possible
by
others
L2
sample space
↳
notations
Evert
↳
:
set of all
s
:
subset of
:
Empty
event
{
=
42,3 }
E
is
,
,
(O )
null event
contained
Properties :
( 0 d) ,
in F
M if EF
EU F
:
law
Associative law
De
Principle of
Some
experiment
=
Hor T , j
=
H or T }
,
{ all
ordering
of A , B , C }
Morgan 's
0 ≤
:
.
PCE )
'
In Ex
.
sample space
:
=
ETF
,
∅
E and F
,
Ed F
law
mutually
EF (EX F)
one in
F
Viki E- i
,
E and F
.
exclusive
are
.
/ Min
#
Ei
Ec
,
identical if ECF
,
E- of
-
-
P (E)
1-
=
are
-
≤1
PCE )
P(UH , Ei )
has
=
=
-
Generalized Form
:
i
:
-
Distributive law
Properties
's )
,
ECF if all outcomes ME
:
Commutative
Axioms of Prob
{( i
an
sample space
=
Union and Intersection
↳
outcomes of
possible
equally
(U ¥ , E) Eni PCEI) if
PCEU F) =P(E) 1- PCF) PLEA F)
,
PCs ) =L
P
,
=
=
,
all
are
mutually exclusive
-
,
=
likely outcomes
:
P {Ei }
=
P{Ez}
=
-
-
=p {En }
=
YN
Good luck
14
Random variable
Discrete
Random
:
var
possible
whose
.
Probability Mass Function
values
Ki }
:
."
i ki
=
-
value ,
a>
:
=
Y
↳ ECXY] Eij (Kisi) (pi Ei )
=
Variance
Properties
Var [× ]
Var (b)
:
X and Y
↳ if
Bernoulli
P(✗
=
K)
P ( ✗= K )
↳
EH]
G. RV
is
K
-
-
(ni p)
→
=
xi
I
"'
-
P)
Yp
p
E [✗
]
2
,
memory
less
:
=
trials
n
,
.
.
if F- ✗
.
(Elly)
2
( 2- p ) /
p2 ,
×>
Vorcax)
,
NOT
p (1) = 0.2
E [✗ it
.
.
.
.
value
.
13=12×0.2 )
TAKE [✗ it b
for ALL X
,
Y
C- [×]
=
=
of Var ( X )
P{ ✗=L} =p P {✗ 03=1 p
Then
=
-
,
var (X) =p
=p ,
-
p
.
2
experiment
.
,
first
E- [×]
,
=
=P( ✗
=
up
independent
one
success
Vor (X )
)
expected
then
,
umps pti)
Vor (X ) 1- Vor ( Y )
Cnz P )
n
µ
,
E= , P(X=k)
.
a,
=
Ki ) and I
E [ Y]
.
k successes
get
,
and
2
.
,
.
-
-
=
a- )
p
.
P(✗ ntkl
=
infinite but countable seq
intervals [Kit
on
AKXK tb ]
✗=L , failure is 11=0
1<=1 , 2,3
,
an
) P(Y=yi )
EH]
Vor (Xt 4)
Number of trials needed to
(
=
and Xzn Binomial
,
Ii )
-
number of successions of Bernoulli
(E) pka Hn
=
=
t
- -
E [X2]
=
In .pt On
:
=
:
,
success is
n
If XY Binomial
Geometric
Y=yi ) =P (✗
-
var (✗ 1- b) = Vor CX)
:
-
,
-
↳ ECXN] of Bernoulli
Binomial
xi
a
P( ✗ ≤a. Y ≤ b) =P ( ✗ ≤ a) P( Yes )
⇒
(Ei Ri pi ) (Ejyjoej )
,
trial
a
constant
is
finite sequence
i)
weighted average of all possible values
:
E [( X µ )2]
=
0
F
,
" ≤•
independent
independent
one
suppose
:
=
=
=
-
:
are
PC ✗
⇒
u
27=1 Pki) =L
ECX] tb , E [A. ✗ it
a
.
↳ discrete
)
→
as
E=ii (e.g
p (ki )
Linearity E [ ctb ]
Independence Ru X and
:
mean
,
Pki )
=
Expectation ( exerted
_
be written
can
ki =P {✗=
p( )
Cummilatine Distribution Function : f- (a)
Lf E[✗]
real numbers
Mapping sample space to
:
denoted
→
then
by
=
npctp)
✗ it ✗ an B
Cnitnz , P )
✗
I
=
( 1=
,
Vor ( X )
,
)
K
p ) / p2
⇒ still need K trials
n
after
fails
.
Notes