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