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Exam # 1
Continuous Random Variables
Independent if PLAIBT-PCAHPCBM-t-plblpcn-nbt-pcn.mn
Two
:
g.gg# The probabability
oÑ*
The
,
can't
pmflpdf
-
.
happen at the sometime (NEVER INDEPENDENT )
mass
function pofa discrete random variable
Flik
pcat-PLK-alfor-ascac.es
bound to
/flxldx
derivative of dist function
.
dist function
.
integral pmflpdf
/ Ill "
median
1m fly)dx=
→
)={
FIX
-
too -11×1=1
interval , make sure to
Of the intervals
-_
✗
=
-
f-
"×
42
-
PlX≤ a)
-5lb )
=
-
Flat
Finding cdfofapmflpdfata specific
Pla≤X≤ b) =/abtcxldx
Quantile :Flqp1=PlX≤ qp / =p
•
Pla < ✗ ≤ b) =P( ✗ ≤b)
Note :
or continuous
probability density function
1-1×1=1--4×1
.
.
.
Disjoint no overlap
Discrete : countable
Plxtakplx≥ a)
events : MAMAN .nAm1= MADMAN .PlAm)
or more
✗
add the full integral
below it Go lower
on each
.
interval
.
✗ ≤ -3
0
44%4-4--21=411--1%4 dx -3*-2
%
" " ≤3
'
Yadx
-z≤x≤z
%%dx+fj%dx2≤x≤
3
I
I
×≥3
#2
EXAM
flxldx 42
FIXKYZ
-_
Median
Discrete
1) Bernoulli distribution Blrlp)
2)
PlX=l/ =p
where
o≤p≤ I
PIX-01=1 p E- [X] =p
Varlxtpltp)
Binomial distribution Binln,p) , where 0≤p≤ I
-
PlX=k)=( 1) pklt-pln-kfork-ql.int/T=npandVarlX)=npll-p)
3) Glow, Where 04pct Geum where 01Pa
4) Poisson distribution Poisltll where µ 0 PLK-kt-%Fe-ntork-o.li
plk-kt.pl/-p1K-'torlh- lR.- ElXI=YpandVarlx1- 1-p1/p2
>
,
FIX]=MVarN=M
,
.
continuous
5) Exponential Dist Expat , where d> 0
6) Normal dist NIMOY where
ounces and
E[X]=M VARIXKOZ
use 2-table
.
"×
flH=Xl
_
FIX)=l
-
e-
7) Paretodist Park), where
.
¥+1
FIX)=tX
fork≥ /
-
"×
E[X]=y× vary)=yµ
for ×≥o
-
.
NO
✗
for
]=✗/L✗- 1)
E- [×
×≥ /
VARIX)=✗ /((d- 1)4×-2)) fort> 2 and
for a > land
8) Uniform distribution Ulaid , where
forocxa
flH=É
as for 0 < ✗≤ I
for
✗ ≤ ✗ ≤b
1=411--5-8
acb
for
a≤x≤ b
E[X]=(a+bk
Varlxtfb
Expectation ? variance
discrete :E[glXD¥glailMX=ai)
ANTINOUS :-[[×]
=/
-
f.[ (✗ +412]=E[✗2)
•
Xflxldx
EEY
as
+
]=[ yflyldy
-
E[Y2]t2E[XY]
COVLXY )=E[XY ] (E[X]E[Y])
-
play)= COVLXY)
varcxivarcy,
E[X+y7=E[X]tE[y]
as
E[X7=f%fWdX
-
WANT ]=E[T2] (ECT])2
-
covariance
correlation coefficient
varlvtwt-varl) vltvarlwlvarlrwt-rzvarlwvarl-YT-varlxtyln.ro
'2=var(X
A-
varies
Jensen's
independent
binomial
2- HEAD
npq
Inequality
g. LEAD 1=[91×1] concave gleam ≥E[gH)]
Dependent if : planB) =/ PLAIPLB)
)≤
convex
the law of
large numbers :
Chebyshev's inequality
:
]=u&VartXn=%
E[In
PAY EEN ≥ a) ≤ atvarly)
-
PHY Ely] / ≤ at ≥
-
2- scores
:
Zn=Tn④-U)
o
m
.­
metrical
0>0
Zn=In-E[In]
Varun
1-
v%#
2n= ✗ it
.
.
.
.tl/n-npl
Orn
as
ECXY
]=y)dydx
a
a.
-
a) 412
suppose that is known that the number of items produced in a factory during a week is a random variable with mean so
a) What can be said about the probability that this week's production will exceed 75?
b) If the variance of a week's production is known to equal 25, then what can be said about the
probability that this week's production will be between 40 & 60?
.
Let X be the number of items that will
a) By Markov's
inequality
PEX>75}
≤
Ef
=
Hence {1×-501<101 }
and
so
the
-
probability
produced
in a week :
5¥ }
=
b) By Chlbshev's
inequality
PE 1×-501≥ to} ≤ ,
≥ I
be
=
4-
Ya
314
that this week's
=
40 and 60 is at least 75
•
.
production will be between