Summary Notes Probability and Statistics 2

DISCRETE DISTRIBUTION
f- ✗ ( N) =P ( ✗ )
-_


Zyfcniy ) =n




fy (y ) PCY )
? f- Cniy ) y
= = -_




INDEPENDENT ⇒ f- ✗ (N ) •
fycy ) = f- Cny )
f-✗ (n)

E- ( X ) = 2- IN
n y
folly ) =

In n§fCn =

In nfxtk)


E( ✗ 2) n' f- Cny )
I§
=




Var ( X ) ECXZ ) [ ECX) ]
'
=
-




=
E- [( X M -

)2 ] =

2- § ( ✗ MY
-



folly )

=


In (✗ -
n )2f×(N )




n!(n?#.-
" ""
(B) ( 1- P× )
Binomial pmf :



n!y!¥¥ 's " -
N -

Y
( 1- Px Py )
"
Trinomial pmf : (B) ( Py ) -




,




qq.az#n-.n.y.zy1.- " " " Y Z
( Py ) ( Pz )ZCtP× Py
- -




(A)
-


"

Quadnomialpmf :
- -

pz )




C# ) PCA ) = 1- P' (A)


Cov ( X Y) E- ( XY ) ECXIECY ) In § ( n Mn ) (Y My ) fcniy )
-
=
= - -

,




correlation , p =




E- ( XY ) nln 1) Pxpy
poxoy § }- nyfcniy )
-
=
M ✗ My + = =




Least square regression line :
y= My +
p 0¥ Cn -
Mx )


Mx = nP×
Var ( X ) =
Mx ( 1- Px )



Independent ⇒ ECUCXIVCY ) ] =
Ecucx ) ] E[ v17 ) ]

⇒ covariance ,
CoV IX. 47=0

⇒ correlation 0
p
=
,




?⃝

, CONDITIONAL DISTRIBUTION ( for discrete ]

h( ✗ / Y ) =fCRiy)_
fy ( y )



E- ( ✗ 14=9 ) =

3- Nh ( NY ) = E- ( hlxly ) )

-

E- ( X / Y=y ) hcxly )
In
'
=




var ( ✗ 14=9 ) ( ✗ 44 ) [ ECXIY ) ]
-
f-
= -




line for ECXIY ) (y )
least
square regression Mx + p M
= -



,




e- -1
Pca < 4lb
/ ✗ =
a) =
I hly In )
{y :acyLb }



CONTINUOUS DISTRIBUTION
f- ✗ (n) fyfcniy ) dy
dy 4414
=




> sketch ration du ⇒
fy ( y ) =L fan ,y , an




Mx = F- (X ) =

In Nf ✗ ( n ) du
0£ fyfnnfcniy ) dndy

( #) if pdf is constant ,
can use ⇒ Volume = Area of triangle ✗ height

independent ⇒ f- Cny ) = fx ( n ) fy (y )


E- ( x2 ) =


In N' f- ✗ ( N ) dn




Cov ( X , Y) =
E- ( XY ) -



ECXIECY )


correlation , p =




Least square regression line :
y= My +
p 0¥ Cn -
Mx )


Varcx ) = F- (x2 ) -


( EG ) ]2