Last
notI
r
-
A Cl
Lisa subset of
A
Catleast
RE
E
e
V
=
M
=
AMB if
REA and
AUB
eEA
if
or
&
EB
of
lEB
AND
=
elements of I not in A
DE
(AUB)"
(A1B)"
=
=
MOURE THEOREM
AC 1 BC
A UBS
and ,
CAMB v(AMB)
=
A3 Important
element
t is not
OR
A = A complement
one
in
A)
PARTITION
1 who
B
A
(Dividing
St
S2
sports)
(A1S) r(A1S)v(AlSS) v
sa
(AMS)
=
>
-
>
-
So , S2 , Se , Su
B
(A)
,
=
A
)
paration I
BC
parkhon &
no
of elements in A
=
CARDINALITY
of A
INFINITY
countably infinite
S. +
=
Uncountably infinite
:
S2 + Sz +
Sit + Szt
......
+
Su
. . . . . . . .
PROBABILITY
AXIOMS
&
set
=
denote
Event
=
P(E)
=
any
our
of possibilities
subset of a
probability of an event
AXIOMS8
1) P(x)
=
1
2) If (A1B)
=
&
(disjoint)
P(AUB)
3)
P(A) + P(B)
PlE)), o
If
4)
=
then ;
A
,
As
,
Az,
......
An
are
pairwise
disjant
then
P(A : 1Aj) =
for all i
,
j
P(Ai) = PLA
then
5)
210
=
a
=
p(and)
P(Q)
6) P(AUB)
=
P(A)
+
=
P(B)
0
-
formulas
ful
P(AC)
=
1
P (AMBC)
P(ACRBC)
P(A1B)
A
B
P(A)
=
P(A)
=
=
-
P(A1B)
PCAUB)<
(Double check)
Day 2
:
Eg : Flip coin 10 hime
IRI
Eg
:
Eg:
210
=
Snuffle
cards
121
52
Pick
:
=
top cand
and look at the
.
random suit
a
121 = 4
Eg
Pick a random value
:
H1 ,
%
=
13
picking
a
random cud
=
=
1 (g)
=
(f )
.
x
4x13
52
& We can multiply different I I
1-2)
PERMUTATIONS
#
of
:
a dishnet ways to order
=
!
n
U
El
~
COMBINATIONS
# of
=>
ways
(v)
?
to choose K elements
=
from
"C =ors !
n
Eg
Draw and
:
from deck
.
Roll 6-sided die
twic
& =
n
+a x +z
x
+
52x6X6
=
&n
Q:
=
P(Draw heart
,
52x6X6
both rolls
woodd)
#x3x3
Eg
:
No
·
of different
#1
=
=
·
(tisu)
# of
flushes posible
ways
to
pick Scand
(5)
:
PlE)
=
If
Eg
E and
then
,
:
P
are
independent
PCEN)
=
PE) X PLF)
(Straight 1 Flush)
:
Straight
Flush
F
- - - - -
Homework /
-
&
2
.
Okan
3)
:
65 +
↑
(April)
300
=
365 + 366
↑
(feb)
28
=
366
.
2
6)
(3)
·
(l-p)3 ph
·
②
Flip
a
coinn
Ans :
In
③
th
=
⑦
181
=
⑤
(2)
↑ (Ace)
=
6
2
X
2"
x
121 =?
times
"
jax12"
52
== i
p(Heat) = =
p) Ace of Heab)
= X
=
se
⑥
Since the two events
for the
#
next
are
cout
1)
=
⑦
14
IE1
the
121
=
51
(we don't put see
first and back)
52x51
52 !
=
26 !
=
not independent
x 26 !
↓
↑
(b)
(brst2real:
P.
..
..
13 ! x13
) =
x73X
⑧
52
12
=
. 10
4
52
PLE) =
5252
Doubt
-
-
P(52 (and exacty1)
=
-
E
&
& p(77)
p(n ++ 4)
↑
⑩
Ex Extxte it
=
:
=
-
to =
6
(HTTT or THTH
or
HHHT)
:
Es itt
=
121
65
=
1
p(A)
↑ (B) :
=
,
2 , 3,
45,6
j
eS
=2
↑
(even)
=
ExExxExt1@3GSE
I
=
ser
#
baking
=
p(BMC)
as
I
p(A1B)
=
0
p(anc)
=
0
P(ABC)
=
q
P(AUB)
j
=
=
&
p(AvC)
p
(BUC)
+
2
+ 3
+
=
P(B) + p(x)
=
=+
p
(AUBUC)
=
-
P(BMC)
E
-
P(A) + O(B) + P(x
Any
-AB) -
p(B1C)
ABC)
-
(maybe
+
+
&
#
=
2
①
people
(2 % )
.
⑫
IE1
P(E)
=
54 ! x99 !
: a
xhl
Day 4
Eg :
4 die's
I wel
distinguishable
,
31
=
,
2
,
....,
>
-
rolled at once
634
Is# from Red die
Eg
:
2nd # from
black
zwd#
green
4th #
blues
What if
you
two
purple
replace
green and blue with
dies .
1st # from Red
2nd # from black
3rdd art #'s from Purple
(rl
= x((2)
2nd
from"six
add 6
possible
numbers
double
choosing 2
not counting
outcome
of
5, 2
= Double
and 2 5
,
but
missing
5, 5 ; G 2
,
etz
Eg
:
Roll 2
purple
dies
unhil it double
p + p2
=
pa + p
1
-
1
·
=
0
I
3 numbers
.
in
outhmells
progression
twice the first tem
Middle term is
Sum
of forms is 120
.
2a
A
*
120
a + 2a + y =
3a + x
=
X=
2a
=
a
a
=
Ga
120
120
-
39
120
=
120 -
=
a
-
3a
-
2a
5a
120
a=
12003a-2a
= 20
①
②
20
⑤
④
40
60
80
3 bicycle riders
all role ten
->
some trace
than second.
⑪
-
2 mils/h
②-
>
second havds 3 m/h faste than
faster
①
②
B
a + 5
a+3
O
#D
②
2
b
hours
①
②
E
b
-
-
~
hous
I
anves
earlier
earlier
than
③
3
Kab
Heid
b
ab
ab
=
=
(a 5)(b 5)
+
ab-Sa
-
+
5b
-
25
5E
-
ab = 5
ab
Ab
=
b
=
a
-
=
(a 3)(b 3)
+
ab
3
-
-
3a + 3b 4
-
Probability equations
Multiple/Recensive
Random Variable
CDF
Inho
-
-
puf
paf
Experience
Expected
values
CRAZY !!
↓
·
Y3
d
the
chance
depending
of
G
=
-
-P
PB = PB +
Pr =
proor
gPB +
Pr
=
1
Pe
=
/P :
not if
e
↑
↳
13
O
=
ho left staching all
depart
P) probability
i
Na
=
1.
=
A, B L
GB
&B = (am)
+ (a) + (0)
,
q
=
(5)an
+
(a
+
5
Sina
Ga
=
= &B 18 ,
+
GB
:
39B
3.
as
= aB
+
33-E)ar =
zaB +
=
82
+
Mi
zr t
Discrete Random Variables
p(X k)
=
()
=
ph(1 p) - "0cXn
-
-
Continuous Random
Y density
of
Valuables
between 10
I
and 12
i
p(10 ->X = (2)
=
E
(Inlegral
↑
CRETE
RV
function
Discrete R VI
.
probabilitie
Sum of
has to be
A
1 +
Set
I
of values
it
can
take
Vi , Va ,Vz
PMF
probability mass funchon -
:
Eg
:
Eg
:
Px(vi)
X
=
1
IT
=
X
#
-
=
of Xand
P(X)
P(X vi)
=
P(Xy0) p(X
=
243
1
wp
=
2
p(X
+
=
1)
i
=
Un
function
X
=
...
E
..P
t
8
=
=
i)
,
1 ,2 3
,
. . .
Bad example
Eg E
:
X= i
go
wp
i
i = 1,
2
2
,
doesn't add up to
probability
5 y
+
one
way
+
:
. . .
== 2
t
it
multiply
>
# add
add
or
=
0 wit
second
a
p
=
/
S a +
S=
S
S
s
=
ar
a
-
+
1
[S]
=
a
=
a
rs
(1 r)
-
av +
v(a
a+
+
+
ar
av
+
av
+
-
I
y
Not importa
=
Random Discrete Variables
E(X)
=
expected value of X
=[p(X
p(X x)
=
-
=
E(X)
=
y+ z
=
of values of X
vi) vi
fork
+
weighted average
=
y
+
=
2
1
,
-
+
,
3
5
4
,
,
+
.
G
.
...
Bernoulli's
X has
dismbution
P(X 1)
=
p(X
+
=
disth
bernoullis
=
0)
even a
--
p
=
=
0- p
with
panametap
o P
+
-
++ +
-
+ +
-
+
+ +
A
+
+
-
-
+ +
-
-
+
-
-
+
-
-
-
+
-
-
-
+ +
-
+
+ + +
+ +
-
+
z
-
+
A
/
-
t
+
-
-
-
t
if P(X X)
PCY
=
E E'l
,
,
=
y)
P(EIE"
=
P(E " (E)
=
independent events
↓
=
8
random vanables
PCE)
P(EP)
=
=
P(E)-P(E")
p
=I
23
F
flip)
a
flip
H
Flip 2
.
independent
Flip Coin 3 times
(2)
are
independent
P(E'RE)
E
#
independent
XY are
=
=
=
fliph
flip3
flips
P(F) =
+ Ext
p(u) =
P
(F(a) =
REPATED
same
+
=
P(F)
.
Exx =
PCA)
BERNOULI'S TRIALS
ALL
with
probability.
&
B (n p)
#
=
,
geomch (p)
=
of successes out
# of trials
3
until
n trials
of
the
first
success.
Even if
failures
Same .
P(X
Ps
.
e
.
are
=
independent
#
=
of thats
untill
B(n p)
x)
.
nth succes
,
=
the
remain
. . . . . . .
n, p
-
the
,
ONE
ANOTHER
X
i
have conhnvors
you
=
(2)
(n)p"(1-pnou
px( -p)
"
=
+
Eg:
Roll die
times
100
P ( more than
p-getting
5)
one
t fp(no 5) P(om3)]
no (0 po(a-py0 (100) pl
S
p(p)
=
+
=
=s
=
-
(10% (5)10
no
n
-
=
(5 )100
-
,
-
%
%
.
S -p
(1 %0) -(8)00
/570
Geometri Examples
Y
~grom (0)
P (Y = 1)
↑ (Y
=
2)
=
=
success at
first
mal
=
p
- D-p)p
failure followedby
p(y
=
3)
=
(1
-
p)2p
FFS
Q pyn +p
-
p(y
=
k) =
-
VALUE
EXPECTED
X
P(X
=
X) given for all values
X
panf
↓
prob
*
PX
n
=
was
x
x
can
that
take
.
p(X X)
=
*
Px
=
EIXT =
X
I
average
value of
the random
variable
=
weighted average
.
Finding expected
Bernoulli
value in
~ Bemoulli (p)
dismbution
Y
=
1
wopp
Y = 0 wop (1
E(y)
E(y)
=
=
1
.
p +
-
p)
0
.
[1
-
p]
P
E(aX
+
BY]
xE[X] + BE[Y]
=
3
Best
X
i
Y
friend
Binomial (n
,
P)
E(X)
=
E(X)
up
=
3
r (n)p"ll -pa
"
.
geomili (pl
Wi
E[w]
up
=
=
Yp
↑ (Y
=
n)
=
p(1 p(
ElW]-&
-
-
nP (W
Epi
E/WT =
BEST
FRIEND
An parnhon
12
E[M]
P(A )
=
.
Y
·
is
an
RV
E[Y1A ] +
.
P(A2)
E[Y1Ac]
·
+
P(As) E[X/As] +. . . .
·
Be event p
00
E[X]
E(y]
P(B)
=
p
=
E[X(B]
.
+
1
.
gom (is)
You
E(X(Y _ ]
E[y]
=
1/ 1/5
=
An = Y14
Az =
happen in first
Y)4
=
As
=
5
+
p(BY
(1 pE(y]
-
even
-
f[y(B]
ECy]
5
=
=
P(Ap)
[1
ETY(An]
.
30
-
STED)
11 p,
-
ECYIA2]
(1 p(0 x)
P(A2)
+
+
↳
·
-
.
2
:
123
Y
=
0
1
n
=
-
-
0
1
Y
9
Y= 2
Y= 3
p(1-p)
p(l-p) pClop)
p(1
+ (
/]
-
p)
+
4 pa + (
-
=
-
p))
p
+
p(1 p) + p(1 p) + p(i -p
-
-
a
.
↑
(0 - p)"
n
-
(
-
-
1
p)
d
P(Y(4)
1
-
(1
-
pj3)
Eigh
E[g(X7] for
E(X] and
~ (8
~
,
V (a
1)
exp(1)
~
,
b)
conhnvor, Ru
dist
uniform
exp(X)
Exponential
dist
CDF of X
pdf
Fx(t)
=
P(X (t)
=
P(X(t)
Ex(t) density
+
du
(s
x
10
du
=
=
Ex (t)
E (
E(X]
Sortylus d
=
-
E(X2]
=
D
Jord
Grid
E(X]
Q:
fxlts
So Bsint
1
- Bcost1"
=
1
(
+
B
,
2, 3 0
,
() pr
oth
O
=
=
Ot M
Boint
"
up
=
Px(v)
=
v=
Binomial
,
+
B]
:
·
/
E(5X]
=
E/5
x
.
]
*
*
5 it d
=
E/3X" log (4 t)]
+
+
d
1."(3t
+
log(4 )) St dt
+ +
·
E[g(1] :"gets facts at
Fy(t)
Yvexp(2)
Y
=
--
1
-
e
~exp(X)E[X] small
lambe
E[Y]
as
gets big
1
=
X
if Fy(t)
=
1
-
e2 +
Let
X
wexp(x)
f y(t) Xeoxt
=
t >0
fx(t) =?
=
Xe
p- def
-
EIW] = wh ear de
e-de-S2 .
o
zw
=
·
-
=
-
2
Jew
Swewo
-
2
-1
~
we'dw
we
2w
-
+
-
In
29
w
-
dw
X ,Y
Indep
X
y
2
=
min
P(z)t)
exp(5)
~
,
=
P(X)t
1x(t)
p(X)t)
P(Y> t)
loat xpo5z
=
lo7t
=
X ,,
min
Exp(2)
3x 17
=
&
~
X2
,,
independe
2mexp(2 + 5)
Independent
(x x2)
d
~
~exp (xi)
exp(xn + x2)
Example :
V time untill
anval
first
~
exp(1)
in hous
Whine
event
w=
unho
first
arral
~exp( ? )
P(v)t)
Gov
P(w) 60t)
=
60t
minute
US t
-
=
t
e
=
p()t)
=
S
in
est
P(w(s)
-
=
e
=
2
-
we
v
v
exp
t
560
(10)
exp(1)
Stats
Oder
.h m
7
Xi
.
iid
exp(x)
z
Y:
indep
Then
i
um
What is
=
,
,
+
,
Y3[t
=
v/n
.
...
+ 1
(xi)
1, . . . , n
Yi
this ??
ES X >,
Yet
()
Yet
P(z (t) =
exp)i)
,
=
Fw()
=
x
,
-
-
=
(Fx(t)
P(y) t)
x
-
,
Fw(t)
)
(1 p(X(t)) (0 p(y(t))
[1 fy (t)][n Fy(t)]
=
p (w( t)
x
1 y (+
p(X (t) + P(y(t)
=
P(Xy +)
=
P(w > +)
p(x - t
=
Min(X , * )
(w >, t)
* t
X
p(2(t)
as
1
=
.
.
& (t)
P
=
~exp
i= ,
independent
Min(X +3
z MAX(X
We
1]
DuBT !
:
If X and Y
W
,
EThth smallest out of n]
Minima Maxima
fact
VT0
[1
1
-
-
-
Ex()] [1
[1
-
-
F + H/]
Fx()][p
-
F-
()]
fy(t))
n
&
fore :
Max&X
2 =
Fz(t)
Ex (t)
=
Fz(t)
&
(2)
,
=
2t
-
=
Bo 13
t
t2
=
Xi -U
Fy(t)
·
to
=
Y
,
%Ede
/2 dz
de
=
+
=
#(2)
=
=
How
to
if
Tw
T
(7
scale these
-
a
U
ab +
b
different
/a b]
,
b
-
a]
,
a
(b a)V
=
+
Vw
(x y)
,
-
~UT0 1]
-
Uniform distributions
E[T] = a
,
V[0
-
-
to
a
Ego
Yn
,
-1/1000 1300]
Y
,
E(max 3Y
,
Y23] = x(3007
Eto
↓
1000
+
-
-
Emportant
-
2/s
>
of the way from
the
starting
since
Xiv(0 0]
,
and
F [man(X, ,
2]
-
Ego
Yi
E[
-
max
V[06 07
i = 1
,
(YoYz]]
:
6+
-+
=
,
=
2
2
2x5
+
=
-
12 + 10
+
3)
E
#
May/maxhYYVery import
a
wit
-
17 i ,
E (max (x1
2)] : &(t]
Ego Yi-VE
=
,
1
+
+
=
2
[min 34.)]
max
=
Fz(t)
=
Elt)
Elt)
- (t)
2
P(X [t)
P(Xa(t)
.
=
=
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prochme Dj U2 07
with
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,
yo
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e
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Twice
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e
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like
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,
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mater)
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Xi
1
=
,
2
,
3
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n
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Some dist
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meanh
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CENTER
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In
=
Y
x
=
THEOREM
= Xi
N(0 1)
~
,
1%
100
8
I
=
sample
mean
2
Date: 11th April
A + B
-
B
Correlation VIs Causation
>
-
TOPICS
A
Common
Cause
Feedback
Loop
-
STABALIZIN A
(f)
&
Fish
Population
-)
Flip Example 5flips
10
Project
(t)
&
Fishing by
Humans
Human Hunger
(Stabalizing)
Balancing Feedback
↑
Loop
If
Human
Population
+
↓
)
the
signs
is
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·
of
ODD
EVEN
201
-
-
human Pop 4
services
of events
cause
it t
go
down
again
Mu
Diseased
Draths
Count
flips
Fair ?
Amount of
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Spurious
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↑
signs
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If #of "
BALANCING
-"
Jedback Loop
UNBANANCING
UNBALANCING
FEEDBACK
LOOP
In order W
Hostile
Anger of
Es speech
of B
A at B
fix
↳
UNBALANCED
Loop
Anger of
& Hoste
B toward A
e
+
FRFRF
Serious
F
of A
&
+
an
is to
change the
way you
to inpirt
react
April18. th
X
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Joint
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