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Final CIS 2033 Cheat Sheet

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random
experiment:
than
more
outcome
possibleof
subset
inclusion-exclusion
the
P(AUB) P(A) P(B)
=
+
P(D) 0B) p(x)
CA, A2 As
Three axioms a
any
&PCS)
pan)
-
rnc)
-
ranne)
A, PCD)
1
=
disguint
are
(notice
a
=
P(ywis) P(A) P(B)
,
+
P(AnB)
principle)
P(A, UA2) P(A.)
them
B
ar
A
for
dand
they
as
P;
P(AB)
PPB
=
PD
MAn)
total
.....
A
is
B
=
P(A(B)
)
LTP:far
B
=
0
Rule:
type problems
add
prob. ap
leading to B
each
"branch"
1A
-
-
Liga
FarPDIB? isickpubrand
Bayes:
P(B)P(AIB)
↳I D
=
-emer
eare
for
obtaining
a
sample
a
size
the from
a
a
set
size
n,
to
of
ways
farm
sample
#
a
acred
sampling
~I replacement
ordered
w/o
sampling
replacement
unordered sampling
w/o replacement
unordered sampling
w/ replacement
repetition
is
entire set is
-o choose
allowed
always
so
an
the
option
Permutations:order
matters
-
(unique numbers/distinctions (
e
k
n
~Pr
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=
derdet(a):nik!
combinations:
stars and bag method
>"the cookie problem"), ***))...
stars
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-
I:
bars
I star)=(**)
ne
in this
I
single branch
·
=
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[An/A, A2, ..., An
probability/Bayes
"decision-tree"
used for
c
P(A.).
=
=
note:
P(n)
=
Law a
0
=
P(AMB) 0,
quick
P(AMB):P(A(PCB),
(probability);
...
(i)
+
thatA I B ) =
as
P(A, MA -
a
is,
and
A
events
chain
rule
P(A) P(B)
=
remember
-
occured
event
*(AM)
thatP(A(B)
⑰B) i
remember
te
probability:odds that
event
happened given another
an
P/B(A)
=
isdanpatentata
principle
conditional
how
to
P(AIB) 0,
=
I
inclusich-exclusion
P(Az(
+
=
,
,
P(AMB)
as
P(B(A) P(B)
=
,
=simplifies
exclusion
P(A(B) P(A)
,
vine-verse
-
intersect?)
-
Sare
PCA):
151
them
S
extension at
0
=
iP A, A2
P(AMB)
=
(inclusion
equally likely
and
probability:
event
P(AUB) P(AIRCB)
-
-
U...UAn]=A,A2... An
I inte
=
event
an
outcomes in
all
Law:
->
For
&
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+
+
If
annal
P(AnB)
-
=
Margen's
De
Get
A
be
an
principle:
->
->
=
possible outcomes
up
experiment
sample space)
event:
-(AuBrc)
4) ded":math
outcome
(S):set
up all
sample space
->
possible
one
case, 3 stars
and 3 bars
Remember tooutfarconstrates a
tract
the
BernoulliTrial: random
experiment
(denoked
-
An
experiment
are
assuming
Bernoulli
a
(a))p"((
f(x)
-
=
I trials,
consists
trials, probability
p)
n
-
outcomes
eses
where
by El
(denoted
canted
possible
two
only
"success" or failure")
either
as
with
successes
of
expressed
be
can
as
K
-#
ways
to
arrange
↑
↑
prob.
↳e
successes
O
exactly
and
successes
not
failures
multinomial theorem;how to divide
~
assuming
m
n,+nc
=
a
different
groups (generalized
where
nz
+
into
distinct
objects
->
of
sizes
are
, Ma, my
3
(r.,ning)
mining;
how
i.e.
to
ways
many
then
separate
groups,
!
-
theorem)
version of binemial
"BENJAMINY
arange
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en
e
rendern
variable:resultfrom
->
probability
Three
random
function from
sample
punction (PMF):describes
mass
far
axioms
of
experiment
each
spaces
at
probabilility
value
to
&
(x(x)
S 0.5iP
Rx
80(x
=
X
=
xyxx)
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i.e.
=
=
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fer
a
flip
coin
IX is the result,
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if
x
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4
=
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PMF:
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E
random variable
a
its
in
real number R
a
x
0
=
1
=
otherwise
0
O:heads
tails)
1:
Distributions
P(X x)
=
Type
Bernoulli(p)
Geometric
Binomial
Pascal
(p)
(n,p)
(m,p)
Hypergeometric
(b,r,k)
Poisson(X)
Range(Rx)
20, 3
E1,2,3,
C0,1,2,
...,
Em, mx1,
3
...
n
...
3
[max(0, k-r), ...,
min
(k,b)3
x =
piPx 1
<- pifx 0
...
3
means
=
-
number
X
=
*
p)X p
-
(1
(*)p"(l-p)n
-
x
2
.
x!
L
x
number a
=
x
-
m
x
number of
=
to
two
groups
expected
x
Bernoullitrials
independent
Bernaull;
le items amongst
(b & v) ap sizes
divide
how
=
m
until the met success
trials
and
*
successes in
independent
respectively
k-x
x
11
independentBernoulli
of
success
trials until 1st
I mi)pr(t-p)
E
O:
failure
=
1
success,
=
-
20,1,2,
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Rx)
(iP
value
number of eventoccurrences in
=
a
given
assumes
E
interval at
&
Events
⑦
occur
Thematical
time/area/value;
indepently
rate
at
if
each
occurrence
is
other
constant
Xm=sederreps
·(p)
in
X, X2,
->
X, +X2+...
then
Bernoull
...,
in
X and
Y
(n,p),
mBinomial
applies
->
random
two
are
↑
2
x+Y
1
=
x,1X2+ ...+ Xn
them
XnBinaial
sit.
2
seen
X-
E[X]
distribution
Bemoulli
Binomial
(p)
X-
and
e n
[ xyP(X xm)
=
E[X]
=
YkERx
var((X])
distribution
Bemoulli
(p)
p(1 p)
-
*Bahasay
up
see
*Seaman
of
type
P
(n,pb
(m, p)
distributions too
Poisson
"Weighted arabiesthe range
of
Pca)(n,p)
=
(m+n, p)
Binamial
=
value:
Expected
X-
(n,p)
Xn=
Binomial
-
variables
=
for Pascal and
a re
type
+
-
"P
m/p
also
can
-
be defined
as
E[X]-(ECX])2
i
np(1 p)
-
3
pretered
war
<C.r.v.)
Continuous Randam Variable:satisfies
properties
->
of x
I
possible
far
② P2X c) 0
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=
Cumulative
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any
in
disjointintervals)
rem*xabl
X
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Function (PDF):p(a2X
Probability Density
min a
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values
=
=
-
Function (CDF):given
PDF
ap
a
f(x) the
CBA
Fx(x)
Pf(x)dX;Pac h) Fx(b)
CDF=PDF, POF=cor
Fa(x) P(x = x)
=
=
=
note:
-
=
-
is
Ex(a)
PDF
X
-
E[x] for
...
continuous
a
iPX
is
c.r.v.
a
·
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E2x?].-(E[X]12
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n)
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fanation of
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them
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=
·
uniform
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time
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otherwise
exponential
(x
-
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rezam
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·
IV
variance
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from table
at
values
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n 0,a2 1)
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event)
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12
o
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=
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ix 0
=
=
otherwise
Fx(x)
=
E
-
1
-
xxifx 0
=
o
cmmice
/
4)
Pxy(X,
jointPME:
=
4
y
y)
=
Pxy(x,
[
PDF:Px(x)
reginal
P(X=x,
=
i)
ar
12y
=
Py(y) [Pxy(x,y)
=
xGRx
at rich, mean
(integrate
over
of
PDF
far
(similar
2
continue
raw.)
cor(XY) 0xy E(XY] -E[X3E[Y]
=
=
shows
->
level
correlation
as
dependence
(XY)=
variables
-
-
PXY
between
range
///
,
sample
mean
expected
of
veriance
standard
(E[X])
sample
dev.
limit
central
Xm
...
(X)=A
value
sample
N
(S2)
veriare
Y
=
=
(Var(X))=E
mean
1111
Er1,13
is
sample
(S):52
sid. der
E
:
Theorem:
way
to standardize
marmal
approximately
distributions
de e
zm:
confidence
68-95-99.7
intervals:remember
(1,2,3
use
z-score
when
is
a
(pop.
far
known:
2-
use
when
t-scare
a
is
IX
Std. dev. (
score
but 5 (sample Siddar)
is
known:
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distribution
+
=
for
untenown
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rale
90%2I
2
x/z
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away
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95%CI
3
2
x
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et
use
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degrees
ap
Preedom n-1) and to
=
at
2I
check for &/2 (i.e.
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=
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