review of probability concepts econ 4550 econometrics memorial university of newfoundland adapted...
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ECON 4550Econometrics Memorial University of Newfoundland
Adapted from Vera Tabakova’s notes
SECOND
B.4.1 Mean, median and mode
For a discrete random variable the expected value is:
1 1 2 2[ ] n nE X x P X x x P X x x P X x
1 1 2 2
1
[ ] ( ) ( ) ( )
( ) ( )
n n
n
i ii x
E X x f x x f x x f x
x f x xf x
Where f is the discrete PDF of x
For a continuous random variable the expected value is:
The mean has a flaw as a measure of the center of a probability distribution in that it can be pulled by extreme values.
E X xf x dx
For a continuous distribution the median of X is the value m such that
In symmetric distributions, like the familiar “bell-shaped curve” of the normal distribution, the mean and median are equal.
The mode is the value of X at which the pdf is highest.
( ) .5P X m P X m
[ ( )] ( ) ( )x
E g X g x f x
E aX aE X
E g X g x f x axf x a xf x aE X
Where g is any function of x, in particular;
E aX b aE X b
1 2 1 2E g X g X E g X E g X
The variance of a discrete or continuous random variable X is the expected value of
2g X X E X
The variance
The variance of a random variable is important in characterizing the
scale of measurement, and the spread of the probability distribution.
Algebraically, letting E(X) = μ,
22 2 2var( ) [ ]X E X E X
The variance of a constant is?
Figure B.3 Distributions with different variances
2var( ) var( )aX b a X
2 2
2 22 2
var( )
var
aX b E aX b E aX b E aX b a b
E a X a E X a X
3
3
4
4
E Xskewness
E Xkurtosis
[ ( , )] ( , ) ( , )x y
E g X Y g x y f x y
( ) ( )E X Y E X E Y
, , ,
, ,
x y x y x y
x y y x x y
E X Y x y f x y xf x y yf x y
x f x y y f x y xf x yf y
E X E Y
( ) ( ) ( )E aX bY c aE X bE Y c
, ,
if and are independent.
x y x y
x y
E XY E g X Y xyf x y xyf x f y
xf x yf y E X E Y X Y
( , ) ( )( )X Yg X Y X Y
Figure B.4 Correlated data
If X and Y are independent random variables then the covariance and
correlation between them are zero. The converse of this relationship
is not true.
cov( , ) XY X Y X YX Y E X Y E XY
cov ,
var( ) var( )XY
X Y
X Y
X Y
Covariance and correlation coefficient
The correlation coefficient is a measure of linear correlation between the variables
Its values range from -1 (perfect negative correlation) and 1 (perfect positive correlation)
cov ,
var( ) var( )XY
X Y
X Y
X Y
Covariance and correlation coefficient
If a and b are constants then:
2 2var var( ) var( ) 2 cov( , )aX bY a X b Y ab X Y
var var( ) var( ) 2cov( , )X Y X Y X Y
var var( ) var( ) 2cov( , )X Y X Y X Y
If a and b are constants then:
varX Y varX varY 2 x y
So:
Why is that? (and of course the same happens for the caseof var(X-Y))
var var( ) var( ) 2cov( , )X Y X Y X Y
If X and Y are independent then:
2 2var var( ) var( )aX bY a X b Y
var var( ) var( )X Y X Y
var var var varX Y Z X Y Z
If X and Y are independent then:
var var var varX Y Z X Y Z
Otherwise this expression would have to include all the doubling of each of the (non-zero) pairwise covariances between variables as summands as well
4
1
1 .1 2 .2 3 .3 4 .4 3 Xx
E X xf x
22
2 2 2 21 3 .1 2 3 .2 3 3 .3 4 3 .4
4 .1 1 .2 0 .3 1 .4
1
X XE X
B.5.1 The Normal Distribution
If X is a normally distributed random variable with mean μ and
variance σ2, it can be symbolized as 2~ , .X N
2
22
1 ( )( ) exp ,
22
xf x x
Figure B.5a Normal Probability Density Functions with Means μ and Variance 1
Figure B.5b Normal Probability Density Functions with Mean 0 and Variance σ2
A standard normal random variable is one that has a normal
probability density function with mean 0 and variance 1.
The cdf for the standardized normal variable Z is
~ (0,1)X
Z N
( ) .z P Z z
[ ]X a a a
P X a P P Z
[ ] 1X a a a
P X a P P Z
[ ]a b b a
P a X b P Z
A weighted sum of normal random variables has a normal
distribution.
21 1 1
22 2 2
~ ,
~ ,
X N
X N
2 2 2 2 21 1 2 2 1 1 2 2 1 1 2 2 1 2 12~ , 2Y YY a X a X N a a a a a a
2 2 2 2
1 2 ( )~m mV Z Z Z
2( )
2( )
[ ]
var[ ] var 2
m
m
E V E m
V m
Figure B.6 The chi-square distribution
A “t” random variable (no upper case) is formed by dividing a
standard normal random variable by the square root of an
independent chi-square random variable, , that has been
divided by its degrees of freedom m.
( )~ m
Zt t
Vm
~ 0,1Z N
2( )~ mV
Figure B.7 The standard normal and t(3) probability density functions
An F random variable is formed by the ratio of two independent chi-
square random variables that have been divided by their degrees of
freedom.
1 2
1 1( , )
2 2
~ m m
V mF F
V m
Figure B.8 The probability density function of an F random variable
Slide B-35Principles of Econometrics, 3rd Edition