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2. Random variables
Introduction Distribution of a random variable Distribution function properties Discrete random variables
Point mass Discrete uniform Bernoulli Binomial Geometric Poisson
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2. Random variables
Continuous random variables Uniform Exponential Normal
Transformations of random variables Bivariate random variables Independent random variables Conditional distributions Expectation of a random variable kth moment
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2. Random variables
Variance Covariance Correlation Expectation of transformed variables Sample mean and sample variance Conditional expectation
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RANDOM VARIABLES
Introduction
Random variables assign a real number to eachoutcome:
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Random variables can be:
Discrete: if it takes at most countably many values (integers). Continuous: if it can take any real number.
)(:
XX
Distribution of a random variable
Distribution function
5RANDOM VARIABLES
)()()( xXPxFxF X
Distribution function properties
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(i) when
(ii) when
(iii) is nondecreasing.
(iv) is right-continuous. when
0)( xF
1)( xF
)(xF
)(xF
x
x
)()( 2121 xFxFxx
)()( 0xFxF 0
0
xxxx
RANDOM VARIABLES
7RANDOM VARIABLES
For a random variable, we define
Probability function
Density function,
depending on wether is either discrete or continuous
Distribution of a random variable
Probability function
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verifies
RANDOM VARIABLES
Distribution of a random variable
)()()( xXPxpxp X
x
xpii
xpi
1)( )(
0)( )(
Probability density function
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)(xf
verifies
1)()(
0)()(
dxxfii
xfi
We have
).(')( and )()( xFxfdttfxFx
RANDOM VARIABLES
Distribution of a random variable
completely determines the distributionof a random variable.
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F
RANDOM VARIABLES
Distribution of a random variable
b
a
bxa
dttf
xp
aFbFbXaP)(
)(
)()()(
Discrete random variables
Point mass
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1)(
aXPX a
axifaxif
xF10
)(
0 a
1--
RANDOM VARIABLES
Discrete uniform
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kik
iXP
kUX
,...,2,11
)(
),...,2,1(
1 2 3 k-1 k1 2 3 k
RANDOM VARIABLES
Discrete random variables
Bernoulli
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pXPpXP
pBX
1)0()1(
),1(
0 1 0 1
p
1-p
1-p
p
RANDOM VARIABLES
Discrete random variables
BinomialSuccesses in n independent Bernoulli trials with success probability p
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)!(!
!
,...,2,1,0)1()(
),(
xnx
n
x
nwith
nxppx
nxXP
pnBX
xnx
RANDOM VARIABLES
Discrete random variables
Geometric
Time of first success in a sequence of independent Bernoulli trials with success probability p
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,...3,2,1)1()()(
1
xppxXPpGX
x
RANDOM VARIABLES
Discrete random variables
Poisson
X expresses the number of “ rare events”
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,...2 ,1 ,0!
)(
0),(
xx
exXP
PXx
RANDOM VARIABLES
Discrete random variables
Uniform
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bxfor
bxaforab
axaxfor
xF
otherwise
bxaforabxfbaUX
1
0
)(
0
1)(],[
a b
f(x)
a b
F(x)
RANDOM VARIABLES
Continuous random variables
Exponential
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01
00)(
00
01
)()exp(
xfore
xforxF
xfor
xforexfX
x
x
0
f(x)
1
F(x)
1/
RANDOM VARIABLES
Continuous random variables
Normal
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0
2
)(exp
2
1)(
),(
2
2
2
2
x
xxf
NX
f(x) F(x)
RANDOM VARIABLES
Continuous random variables
Properties of normal distribution
(i) standard normal
(ii)
(iii) independent i=1,2,...,n
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)1,0(NX
),()1,0( 2 NZNZ
),( 2iii NX
),( 2 n
ii
n
ii
ii NX
RANDOM VARIABLES
Continuous random variables
Transformations of random variables
X random variable with ;
Y = r(x); distribution of Y ?
r(•) is one-to-one; r -1(•).
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XF
RANDOM VARIABLES
dyyrd
XXdyd
Y
XY
XY
yrfyrFyf
yrpyrXPyXrPyYPyp
yrFyrXPyXrPyYPyF
)(11
11
11
1
))(())(()(
))(())(())(()()(
))(())(())(()()(
(X,Y) random variables;
If (X,Y) is a discrete random variable
If (X,Y) is continuous random variable
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,
probability joint function( , )
( , ) 0
( , ) 1x y
p x y
verifies : p x y
p x y
probability density joint function( , )
( , ) 0
( , ) 1
f x y
verifies : f x y
f x y dxdy
RANDOM VARIABLES
Bivariate random variables
The marginal probability functions for X and Y are:
23RANDOM VARIABLES
Bivariate random variables
For continuous random variables, the marginaldensities for X and Y are:
xY
yX
yxpyp
yxpxp
),()(
),()(
dxyxfyf
dyyxfxf
Y
X
),()(
),()(
Independent random variables
Two random variables X and Y are independent ifand only if:
for all values x and y.
24RANDOM VARIABLES
( , ) ( ) ( )
( , ) ( ) ( ),
X Y
X Y
p x y p x p y
f x y f x f y
Conditional distributions
Discrete variables
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If X and Y are independent:
Continuous variables
RANDOM VARIABLES
)(
),()|()|(
yp
yxpyYxXPyxp
)(
),()|(
yf
yxfyxf
)()|(
)()|(
xfyxf
xpyxp
Expectation of a random variable
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Properties:
(i)
(ii) If are independent then:
niXEXEi
iii
ii ,...,1
niX i ,...,1,
i i
ii EXXE
RANDOM VARIABLES
dxxxfEX
xxpEX
X
xX
)(
)(
Moment of order k
27RANDOM VARIABLES
dxxfxEX
xpxEX
kk
x
kk
)(
)(
Variance
Given X with :
standard deviation
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22 )( XEVX X
EX
2/12 ))(( XEVXX
RANDOM VARIABLES
Variance
Properties:
(i)
(ii) If are independent then
(iii)
(iv)
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)()( 2 XVabaXV
i
iii
ii XVaXaV )()( 2iX
22 )(EXEXVX
0VX0 ( ) 1VX P X a
RANDOM VARIABLES
Covariance
X and Y random variables;
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))((),( EYYEXXEYXCov
RANDOM VARIABLES
EXEYEXYYXCov ),(
Properties
(i) If X, Y are independent then
(ii)
(iii) V(X + Y) = V(X) + V(Y) + 2cov(X,Y)
V(X - Y) = V(X) + V(Y) - 2cov(X,Y)
cov( , ) 0X Y
Correlation
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VYVX
YXCovYX
),(),(
RANDOM VARIABLES
X and Y random variables;
32RANDOM VARIABLES
Correlation
Properties
(i)
(ii) If X and Y are independent then
(iii)
1),(1 YX
0),( YX
baXYaYXbaXYaYX
:01),(:01),(
Expectation of transformed variables
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( );Y r X
RANDOM VARIABLES
dxxfxrXEr
xpxrXEr
X
xX
)()()(
)()()(
Sample mean and sample variance
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Sample mean
Sample variance
RANDOM VARIABLES
i
iXn
XEX1
i
i XXn
SXV 22 )(1
1)(
Properties
X random variable; i. i. d. sample,
Then:
(i)
(ii)
(iii)
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;, 2 VXEXnXX ,...,1
XE
nXV
2
22 ES
RANDOM VARIABLES
Sample mean and sample variance
Conditional expectation
X and Y are random variables;Then:
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Properties:
EXYXEE )|(
RANDOM VARIABLES
.| yYX
dxyxfxyYXE
yYxpxyYXEx
)|()|(
)|()|(
