Probability Foundations
Mutual
Exclusivity
together (for any events)
> no 1 can occur
-
Independence
>
-
A1B = @
:
P(AnB)
:
=
P(A)P(B)
mutual independence requires intersection = Hall individual
PLAUBUC) only requires at least) to occur
Combinatories
pere
ok people
Khoosing
combinatione :
Runderdementatomn
splitting group size n into 2 teams /exactly)
ways needto avoid double count
total distinct
no
replacement
probabilities change > use hypgeom or combinatorial counting
-
:
RVS
pdf total area
=
Discrete cdf
:
1
f(x)
[p(k)
=
continuous cdf F(x)
:
↑(a = X = b)
=
-
>
R ≤ JC
F(b)
-
add pouf of
every value at or below sa
/of (t)dt
=
F(a)
Discrete distributions
Binomial : successes in in Bernoulli trials
X-Bin(n , p) P(X B)
Exx)
( E)
pR(1-pin-k
=
=
=
up var(x)
=
up(l-p)
Hypergeometric selectn without replacement from population sizeN , type I,
:
wtype
2
XNHLW , Rin ) PLX=
R
)=
(
て
λ= VarCX
=
Geometric #trials to get 1st success at 1th trial
=
(1 p)k
P(X k)
p (x)
p2
1
1 pJR
FXLR]
:
=
=
=
-
-
=
=
-
+var(X)
=
rth success occurs at Rth trial with P(success) P
Negative binomial
k
남
스
=
:
P ( X = R)
>
-
=
( : ) ( 1 p)
-
equivalent
P(X
k)
=
γ
p
.
.
t
(x) =
γ
p
to 5 of i i d
Prisson : #occurrences
=
-
geometric RVs
of event within period where X is average
ceXREX
=
X
,
varIX =
npl
Continuous Distributions
Follow Poisson process with rate x
,
Exponential
fy
-
=
=
*
a
x)
waiting time rthevent y-Ga(r
1
E(y)
Gamma :
fy
time to next event
ng E(4)
Var(4) =+
wait
2e xy
:
to
,
= yr
1
-
-
e
y,
yr
=
xy
var (4)
=
λ
=
*
- te xy
-
0
↳ 「 ir ) = S gr
-
-
e
Ir-11 !
=
caf : Plx7x
tinding
]
or convect
↑ (X > x)
=
=
ydy
e-
xe 。
to Poisson ,
P(Poisson (xx) = r -1)
µ
合
exp( - z ( Ʃµ p )
a
uniform fx =
E(X) =, var() =
Normali f 2
Z
=
=
:
,
a
Normal approximation
Binomial
poisson : M
:
=
M upz 02
=
x
j
=
x
=
np(l-p)
up
correction for discrete to continuous
PlEca + 0 5 )
PCX ≤ a)
P(z > a 0 5)
P (X = a)
> 5
,
n/l-p1 > 5
:
=
.
=
-
.
EX/COV/ vAR
For DRV,
E(X)
Forcont
,
Elg((())
[
=
r X
allk
=
=
( @ g(x) +
.
Elg(x , y()
Elaxtby ]
* for any nonlinear function of
py(k)
=
(x) dos
y
variables find new variable's density
B
,
integrate variable x density
((g(x y) fx y dxdy
,
,
at ( x ) tbElY
CoV(X
E(XY) EIX)ELY)
, y1
Cou(aX + b
, cy + d) acCov(X, Y)
var caxt by )
azvarlyx ) t bavarty ) 2ab Cov ( × , 4 )
coucaxtby , . * tdy )
acVar ( X) tbdvarcu , - cadt bcl cov ( X , y )
for independent (1-way) E(XY) E(X(34)
0
cov(X, y)
=
-
=
=
=
=
:
=
correlation coefficient px , y
riXY
=
{
\
ー
1
ㅇ
perfect positive linear relationship
perfect negative
uncorrelated
MGFS
Mx ( t )
t ( et
=
*
)
) , mwlr ( 0 ]
E ( Wr )
=
=
Max + b( + )
Normal
=
Geometric
=
:
if RV has
associated distribution
o
eb+ my(at)
ent+2
:
drmnl
of
mugf
known form it must follow
Uniqueness
Bin(n , p)
et plet
:
=
ll
p
-
+
petyh
Poisson ex1et-1)
:
Negbin
petplet]
I
:
Gamma
(1-El-"
:
r
( **)
or
Exponential :
Order Statistics
largest fin
:
=
n[Fy] "fy
*
n/T-FyJi ty
! IFy ] ' [T Fu ]
i thici
-
smallest fyi =
:
-'
n
-
-
-
Higher order moments
-
ify
oth moment about
n
Mr ECwr)
orig
rth moment about mean EW-Mr]
:
=
:
Existence of moment
:
Syrflyldy must converge (be finite)
:.
rc-1
-M13]
YzW-M
Skewness
Kurtosis
CLT
iid RVs with mean
MB var22 , for su
nu var(sn)
E-M
Elsn)
=
=
=
X , + X2 +...
+
Xn
ngz
Gun
independent normal Rvs
:
X+ Y
~N(Mx My
random sample sizen from N(M , 821 ,
+
,
03
+
24)
sample mean-N(M ,
weighted sum Y
Y
=
a
〜
,
Y , +... + anYn
,
NCYaili , ai oy
2
Point Estimation
MLE (10)
:
= flyis) find that gives
o
,
boundary/non-differentiable
Set ists theoretical moments
MOM :
)
て
X~ =
⽅
; = xir
:
max
order statistics
equal to corresponding sample moment
properties of Estimators
EfficiencyRe
)
rloi
=
.)
acmears move efficient
.
)(
Bas(0) E10) O- > unbiased it 0
ECOn)
Asymptotically unbiased him
c
=
=
-
:
n
CRLB
ー
for
=
0
>
-
IMVUE)
any unbiased estimator , support independent ofO
var(0){-nE[10g +
Confidence Intervals
σ rkwwn
Y±
:
o unknown
:
(4 :
"error in both directions" = &
正圣 ⇌
th i
sample size determination
n=
r" or
E
z2p(l-p)
d2
@]3
2
PEENPII
Algebraic results
differentiation PR U'v + uv'
QR VU'-UV
y
:
=
ax
= axna
:
_
㎡
Joint distributions
discrete
Tmf EEPx ,
in esc
,
probabilities
-
>
marginals
1
=
y1dxdy
:
draw
uniform fxy
=
1
=
Erowsorcol s
=
marginals fx(x)
=
(fx y(x y)dxdy
,
,
↳ bounds are allowed y
regions
cared
region
onditional distributions
-
fyix = xly
)
=
PCXsaly
=
b)
1
-
find f × ly
=
、
y
( ox)
.
2
sub
.
3
integrate over a
my
=
b
Independencefy
Transformations
continuous linear
CDF method :
:
Fy (y)
a
fy = + x()
=
P(g(X) < y) > express in termsof *x > differentiate
-
summing independent RVs
for co
fw
=
,
a 20
BS
5 on fac , lscilfcz (
=
W
-( >
-
-x ,
) doc .
WCS ,
-