Chapter 3 Properties of

Random Variables

Moments and Expectation

Review Experiment Random Variable Observation Realization Parameter Sample Statistic

Moments One way to quantify the location and some

measures of the shape of the pdf.

xdAd '1

AxdA'

1

First moment about the origin

dxxpdA X )(

dxxxpX )('1

ith moment about the origin

dxxpx X

ii )('

j jX

iji xfx )('

continuous random variable

discrete random variable

ith central moment about the mean, m

dxxpx X

ii )()(

Expected value of a random variable X

dxxxpXE X )()(

j jXj xfxXE )()(

X continuous

X discrete

Expected value of function of X, g(X)

dxxpxgXgE X )()()]([

j jXj xfxgXgE )()()]([

X continuous

X discrete

Expected value and the first moment about the origin

dxxpx X

ii )('

dxxxpXE X )()(

Comparing

to

You can see that the expected value of the random value x is the first moment about the origin.

Rules for finding expected values

ccE )(

)]([)]([)]()([

)]([)]([

2121 XgEXgEXgXgE

XgcEXcgE

Measures of central tendency Arithmetic mean Geometric mean Median Mode Weighted Mean

Mean, mx, or average value

'1)( XEx

n

i

i

n

xx

1

k

iiinxn

x1

1

Mean of a r.v. X is its expected value.

Sample estimate is the arithmetic average.

Arithmetic mean of grouped data (k is number of groups, n is total number of observations, ni is the number of observations in group i, xi is the class mark of the ith group.

Geometric mean Used when the ratio of two consecutive

observations is either constant or nearly constant.

n

i

niG xX

1

1)( n

n

i i xxxxxx 43211

n

xx

n

ii

G

1

log)log(

The logarithm of the population geometric mean would be the expected value of the logarithm of X.

Median, Xmd

The observation such that half of the values in the sample lie on either side of Xmd. The median may not exist.

Population median, mmd would be the value satisfying:

md

dxxpX

5.0)(

p

iiX xf

1

5.0)(

X continuous

X discrete

Mode, mmo

Most frequently occurring value. The sample or population may have none, one or more than one mode.

Population mode is the value of X maximizing px(x).

0)(

dx

xdpX0

)(2

2

dx

xpd X

)(1 iXni xfMax

X continuous

X discrete

Weighted mean Used for describing the central tendancy

of grouped data.

k

i i

k

i ii

ww

xwx

1

1

Measures of Dispersion Measures of the spread of the data

Range Variance

Range Difference between the largest and smallest

sample values. For a population this interval often ranges from - ∞ to ∞

or from 0 to ∞. The sample range is a function of only 2 of the sample

values, but does convey some idea of the spread of the data.

Disadvantage of range: does not reflect frequency or magnitude of values that deviate from the mean.

Occasionally use the relative range

Relative range =

X

xx lu )(

Variance, s2

Defined as the second moment about the mean.

The average squared deviation from the mean. For a discrete population of size n:

Sample estimate of sx2 is sx

2

)()(])[()( 2222

2 XEXEXEXVar

n

xi i

X

22

)(

11

)(

1

)(22

222

2

n

xnx

nn

xx

n

xxs i i

i ii i

i ix

Variance Two basic differences between population

and sample variance. used instead of m n-1 is used as the denominator rather than n to

avoid a biased estimate for sx2

Variance of grouped data

x

k

iii

X n

xxns

1

22

1

)(

Rules for finding the Variance

)()(

)()(

0)(

2

2

XVarbbXaVar

XVarccXVar

cVar

Units of Variance Units of the variance are the same as units

on X2. Units on its positive square root, the

standard deviation, sx, are the same as the units of the random variable, X.

A dimensionless measure of dispersion is the coefficient of variation, Cv.

x

sc Xv

Measures of Symmetry Many distributions are not symmetrical Tailing off to the right or the left is skewing

the distribution. Tailing to the right-positively skewed Tailing to the left-negatively skewed

Skewness 3rd moment about the mean

dxxpxskewness X )()( 3

Practical measurements of skewness One measure of absolute skew is to

measure the difference between the mean and the mode.

Not meaningful for comparison sake because it is dependent on units of measure.

Pearson’s first coefficient of skewness Relative measure of skewness more useful

for comparison.

mo

X

mo

S

xx

Population skewness

Sample skewness

Measures of Peakedness (Flatness) Kurtosis refers to the extent of peakedness

of a probability distribution in comparison to the normal distribution.

Kurtosis is the 4th moment about the mean.

Calculate the coefficient of kurtosis, k

dxxpx X )()( 4

4

Kurtosis

Covariance Measure of the linear relationship between

two jointly distributed random variables, X and Y.

Covariance is the 1,1 central moment

)()()()])([(),()( 11 YEXEXYEYXEYXCovYXE YX

dxdyyxpyxYXCov YXyx ),())((),( ,

Covariance If X and Y are independent:

Sample statistic:

0),( , YXYXCov

)1(

))((1

,

n

yyxxs

n

iii

YX

Correlation Coefficient Normalized covariance

If X and Y are independent

YX

YXYX

,

,

11 , YX

0, YX

YX

YXYX ss

sr ,,

11 , YXr

Correlation Coefficient Measure of how two variables vary together. A value of r equal to positive one implies that X

and Y are perfectly related by Y=a+bX. Positive values indicate large (small) values of X

tend to be paired with large (small) values of Y. Negative values indicate large (small) values of X

tend to be paired with small (large) values of Y. Two values are uncorrelated ONLY IF r(x,y)=0. Correlation does NOT equal cause and effect.

Correlation coefficient: Linear dependence and functional dependence

Other Properties of Moments

)()()()( YbEXaEbYaXEZE

)()()()( 22 bYaXEbYaXEbYaXVarZVar

),(2)()()( 22 YXabCovYVarbXVaraZVar