Final_1.pdf

12 BACKGROUND MATERIAL

APPENDIX

Below are a few formulas on random variables that we are likely to en-
counter throughout the book.

DEFINITIONS

LetX,Y, and Zbe random variables. Let (x 1 ,y 1 ,z 1 ),(x 2 ,y 2 ,z 2 ),...,(xN,yN,
zN) be Nrealization 3-tuples for these random variables.

Mean

The mean or expected value of Xis denoted by E[X] = mx.

The estimated value of the mean of a random variable is known as the

average.

The formula for the average is.

Variance

The variance of Xis.

The estimated value of the square root of variance is the familiar stan-

dard deviation.

Its value is calculated using the formula.

Covariance

The covariance between XandYis denoted as

.

An estimation of the covariance may be calculated using the formula

.

Correlation

The correlation between XandYis

The formula for the estimate of correlation is given as

1

1

N ii

i

N

xx yy

XY

()()−−

()()

=

∑ avg avg

stddev stddev

corr( , )

cov( , )

var( ) var( )

XY

XY

XY

=

1

1

N ii

i

N

()()xx yy−−

=

∑ avg avg

cov( , )XY=−E x[]()μμxy()y−

xxxN i

i

N

stddev=−avg

=

∑

1 2

1

()

var( )XEx=−()x







μ

2

xxN i

i

N

avg=

=

∑

1

1