Cheat sheet for second midterm in Statistics for Economics (ECON 30330) at University of Notre Dame. Covers topics such as discrete and continuous probability distribution, types of distributions, and linear combinations of random variables.

1. .
PDF :f( × )=P(X=× ) -
hypergeometric
.
Uniformdistccon .
)
for all x describes how distribution
-
ECX )=µ= #
the distribution of x
-
n identical trials ,
2 -
vary )=oz=
#
2
looks like Outcomes ,
not indepen
.
12
-
Efcx )=1 dent ,
PCs ) changes ( no
.
normal dist
.
CDF : FCXKPCXEX ) replacement )
-
Symmetric ,
means
-
ELX )=EXf(×)=M -
p(×=×)=rCx
.
N -
rcn -
× median -
mode ,
.
Vary )=O2=§(×
-
MPFCX ) ncn larger Std → flatter
.
Std ( X )=o=Fz Where N :# of elements curve ,
tailsextendto
Discrete r :# of successes Infinity ,
total area -1
-
uniform distribution n :# of trials -68 .tw/in+O
-
each value has equal X :# of successes inntrials
;
95% wlin 1=26
probability
-
ECX )=M= F- =np
99
.tw/in+3G-PDF:fCx-n
-
vary )=n
.IN .
¥
Na
-
Pdf :-.
-
k¥2
-
ECX)=§ 't - -
N -
1
Oft
.
e
-
Var ( × )=§ ( ×
-
µ )2 's p 1- p
-
ECXKM
.
binomial distribution Continuous -
Vary
)=02-
a fixed number not .P(a<_×<_ b) =fbaf(×)d× -
X~N ( µ ,
02 )
identical trials resulting =FCb ) -
Fla )
.
Standard normal dist .
in success or failure ,& .PL/l=x1=O
-
14=0 ,
02=1
-
I
P (5) does not
vary from
.
FCX )=P( XEX ) -
f(×)= -1 2
trial to trial ,
independent .pl/lEa)=f?afCx)dx Fit
'
E
-
( n×=( F) =
Finn , , .×li→mos FCXKO -
f(z)=0( Z )
-
PDF : P ( X=×)=f( × )= .
S -%f( × )d×=1 -
F( a) =p(z<_ a) =
Cnx p×( 1- p )n
.
×
.EC/l)=f!8xf(x)dxfIo0Lz)dz=I0(
a )
-
CDF :P(×<_ a) =
.
Var ( × )=fFs ( x -
M )2fCx)dx -
Plzza )=1 -
Iola )
×&o( 2) p×qn
.
×
.
pdf :f(×)=±I¥'=F' ( × )
-
Plazzzb )=Iob -
Ioa
-
ELX
)=µ=np
.
cdf :
FC a) =p(×< a ) -2=1 'T
-
Vary )=o2=np( tp )=
.
uniform distribution Exponential prob .
npq
-
all intervals wleaual
-
time blwoccurrances
Poisson distribution length are
equally likely of some event
-
# of occurrences
-
a continuous RVIS -
Pdf
:f(×)={÷nE¥forxI0Over a specified interval uniformly distiftheprob .
0 elsewhere
oftimebpace is proportional to the inverse
-
FW=P(X<_ c) =1 -
EE
-
P(X=x)=5YTe
' ×
Of the interval 's length
-
ELXKM ,Var(×)=M2
Where X :
expected value -
f(×)={b÷a for aE×Eb jltenohaforhunwybearpootssaomniaatfriwbiutthionn?-
ECX ) =
× 0 elsewhere then the length of time between
arrivals must follow an exponential
-
Vary )=X
-
Cdf :F( C) =P( XEC )= distribution
scaaidx= ¥
Linear Combo of RV Linear Combo of Constant ERV Linear Combo of Constant } ZRVS Correlation blw ZRVS
X :
random variable 4=130+13 , X Y= 130+131×+132 Z
g×y=
Covl X. Y )
C : Constant ECX )=µ× ,
Var( X )=0×2 ECX )=M× ,
Varcx )=O×2 stdk )std( Y ) =¥%y
E (c) =C My =
ECY )= 130+13 , F- ( X ) ECZ )=µz,
Varcz)=q2
°
population covariance :
E ( C. g( × ) )=CE(g(× ) ) 0y2=Var( Y )= 13,20×2 ECY )=Bo+ 13,14×+132 Mz o×y=
EL Xi
-
Mx )( Yi -
My )
ECFCX )+g(× ) )=E(f( × ) )+E(g( X )) 0y2=B,2Var(X)+Bz2Var(z)+2B,BzE[ a- Mxxz
-
M⇒ ]
N
If Y=X+2 ,
ELYKELX ) +ECKES °
sample covariance :
VARLY )=Var(X+Z)=Var(XHVar(z)+2cov( x. z ) s×y=
EC Xi -
F) ( Yi
-
J )
h -
1