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Summary Formulari Probabilitat i Estadística

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  • Course
  • Probabilidad y Estadística
  • Institution
  • Universitat Politécnica De Cataluña

Formulari de temari de probabilitat i estadística. Inclou les fórmules dels models probabilistics, esperances, recurrències, cadenes de markov, estadística, Rstudio i més. Temari en diversos idiomes. Perfecte per preparació d’examens i per practicar exercicis.

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Document information

  • September 10, 2023
  • 3
  • 2022/2023
  • Summary

Subjects

  • probability
  • markov
  • statistics
  • engineering
  • ingenieria
  • data
  • datos
  • formulario
  • discrete
  • binomial
  • hypergeometric
  • moment
  • gamma
  • beta
  • uniform
  • expectation
  • function
  • chain
  • recurrence
  • distribution
  • station
  • summary

Written for

  • Universitat Politécnica de Cataluña
  • Ciencia e Ingeniería de Datos
  • Probabilidad y Estadística
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I* DOM VARIABLES
:



!Fe(X) = 0 (2) Fx non-decreasing f (3) E is right-cont .:
fe(x) - i
Density func
- -




b) cythen Fe(x = Fx() lim
aFa(x) Fx(a)
Fe(x) Distribut
=




↑(ac Fe(b) Fe(a) Prob
Fa(x)
x
=




1 x
+ -
=
- .




function
↑(A = a) =

Fa(c) -
4 Fx(a)
↑(A >x) = 1 -
P(Acx) =
1 -


x
- Fx(t) P(A >x) =
1 -

/A ex) =
1 -


Fe(t)

DISCRETE PROBABILITY MODELS DISCRETE EXPECT .
& MOMENTS CONT PROBABILITY MODELS

·
BERNOULLI MODEL -D noves 2
po- EXPECT E( E ) x PP(A =
x) ·
UNIFORM (CONT) -> toti els
=
.

-
·
.
xim

ssibles resultats 1)
intervals amd la meteixa llargada teven
(0 : la mateixa
probabilitat .




G
IP(A 0)
=
1 - p
P(X 1) P 1xb -
a) x + [a b]
baia x(x)
= =
=
,




f((x)
=


=




probability of success)
,

[a
Sp -x 0 x , b]




= Ya
X La
* =
Ha we wile -Be(p) a
acxcb
, x -




FE(x)
b
* ~u([a b))
x ,



· BINOMIAL MODEL-# d'exits en ma




seq, de Berwoullis indes .

· EXPONENTIAL -

per
modelar temps
d'un event
P(A =

k) =



(2)pk qn-
1
n =
0
,
1
....,
d'espera per
l'ocurrencia
,
->
E(E +
1) =

E(X) +
E[4] -
xx
Newt . # ~ Bin (n
p) e[a bAT bE[A) f((x) =
xe 11 a(X)
XE(A)
-
,
a (
E(XE)
=
+
,
B in
+




k
=




(k)p"(1
U

4))"
->
~xx
,
,


y(4 (p
-




-D
E P( n) = (1 1
1 xedt for x30
=




* 1
= +
- =

e
-




E[E 7] ETAJE [1] INDEPDo Fe(x)
= -
=
:
-
.
.

-
!



x
E(7) E(q(A)) , (
Exp(x)
=

*
=
=




· POISSON MODEL - # d'events casuals

que ocurreixen en u tems fixe -
MARKOU-INEG
:



P(Asa)
= El · if En ~Geom() P(*n :

= kn) =

1-11 -


2) "
- xk

Eso
-> 1
e 1 e
-




4)
Pla
=




x k/P(A x)
=
=




Excel
=




MOMENTS -DE[A"]
=




·
· IP(Ast) * > s) =

PP(A)t s) -




* 40(X) (X freq ,
#d'events milja) · k-th central moment of A : · Poison Point Process :

↑(T1H)= IP (*
+? 1)
x
( A)) "P(A (TrExp(X))
-




↳ m) Polnoml E((I E( *))") z(x x) =1
4(Aq 0) 1 e
= -
-




A ~Bin(n
=
=

m
-




quan nom
- =

=
,
x t (r)
constant (x)
convergeix a ma
modelar
· VARIANCE -P second central moment · NORMAL-D pernet nombroses

ferimens naturals socials i
psicologics
,



6 Var(A) El(A E(Al))
=
=




GEOMETRIC
-




MODEL-D # d'intents
=




· L

de Bernoullis per aconseguir el ser exit .
Var( A)
=



E(12) -
E(A)2 fx(x)
1
qk k 1 2 3 Stand Deriv -D 6 == +v Var(A)
N(m 6) = variation)
-




P(E k) (m expect
=
·

p *
=
, , .. .


~
.
= =




(
,
, ,


·
Chebysher Freg -- P(1X- E(X)1a) -




Normal Distrib >N 10 11
·
SUM OF A GEOMETRIC SERIES ~D · Stand . .
-

,




p(k
: En e*
-"
es en e
2

42 4)k
E n) /1
(1
-




(*
=
=
- -
=
=




,
,




agreen estam
P a
=
1 S
1 -
x -
p)↳
( .
H en



· SLIGHT VARIATION
-
# d'intents falliti ar
oMOIURE-LAPLACE :
Binomial tends to a


del der exit : Normal for
large
ser
baws n -



P(E k) qkp k 0 1

(2) par
2
fa(h)
= =
=




a)
, ,
...




P (A
, =

= ~



· LACK OF MEM .
-
I (A ? r+s1A2r) =
P(A3s) ↳Var(A) = 0 -0
A =


E(A)

↳Var(a) 0 · GAMMA -D
procediment acd
Geom(p)
=



a passos
*

~


(a1) a2 Var ( A) cadasom
↳ var indep pren Exp/N quant temps
=




, . de

(A+ )
/t
NEG BiN MODEL-D de ↳ var var(A) +Var(4) +2 Cor(A 7) ↑ (x)
*
t for
en ma
seq
=
=


· . . .
,
at x 0

el # d'intents fins (E 1) Var(1) +Var(I) ~D INDEP
I
indep ↳Var
a- =




Bernoultis
+




,



rribar a (rexits .
↳ var (I) =



E(var(E11)) +
VarIE/IlE)) ↑(x 1)
+
=


xi(x)T(n 1) +
=

n !
xx
(r 1) pqk-r f(x) x - for
-




k=r 1 (M6F) x30
P(A =

h) =
-
,
r +

...
o MOMENT GENERATING FUNCTION =




et (A x) Ganma(x X)
*




r) E (et ) z
NegBin(p
*
*


~
Louk # dintents) MA(t)
=

= =
:

. ,
x= (2)
E(E4)tk ·
Exp Gamma(1 X) Neg Bin cont version
obtenin la distrib
=




2
=



Me(t)
, .




Quen 1
=




grom
· = .




k!
.



k, 0




· BETA-D Euler Beta fus :
· HYPER GEOM . MODEL -
extraiem mos-



/! (1 t)Y
-




tres de tamany'r' d'una pobl .
detanary B(x y) ,
=


t
*
- d+ for x , y 0



inquete im individus de tipus 1 :
(x)T (y)
↑



(n -m) de tipus 2 Volew comptar els +1
B (x
-

y)
=




, T(x +


y)




e est,
(2)(2 z) =


1 2
P(A
=
0
k)
...

= = , ,


(2)
Hypbeom (n
*



~
,
m
,
r)



tots intervali de
· UNIFORM MODEL -D ell
CONT RANDOM VARIABLE
*



~ Beta(x , !
mateixa mateixa
.

la
la long teven prob .




(" ftd probdesie
.




IP(A =

k) =
Yu k = 1
,
2
, . . . ,
·
Fa(x)
=




CONT .
EXPECT .
& MOMENTS

- -
u(n) ETAJ =
E ,
X:
VII) =
x
FTC :
fx(x) =




Fx(x) 0
Expect - E(A) =
(-8xfx(x)dx
oP( A x) = =



Fa(x) -



(Fx(t)
= 0 0 k-th moment- -(A4) =




(x " fa(x) dx
h) central k-th --
E(1 E(X(4) (-8(x- (A)"f((x) =




↑(ac(a
mom
-
.

o

f(a)
+



=

02nd central mom .
is the variance of A :




6 Var(E) (EY)
2

* E
/ E (A)
= =
-




* [A] =




(/z xyf(x y)dxdy ,

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