Elementary Probability Theory - Digital Communication: Principles and System Modelling - page 208

with elements s, we assign to every s a real number X(s) according to some unique
mapping and X(s) is called as random variable.
The random variable considered being a function or mapping that maps all the
elements of the sample space into points on the real line. The following example
can make our understanding clear.
Figure A.1 shows an experiment where the pointer on a wheel of chance is spun.
The possible outcomes are the numbers from 0 to 12 marked on the wheel. The
sample space consists of the numbers is the set

{ }

0 < s ≤ 12 . We define a random

variable by the function

s 2 X = X ( s ) = (A.2)

Points in S now map onto the real line as the set { 0 < s ≤ 144 } .

Fig. A.1 Random variable as
a mapping/function of a
particular experiment

12

Spin

X

9 3

36

6

9
0

A.3 Mean, Variance, Skew-ness and Kurtosis

The mean [3] of a data set is simply the arithmetic average of the values in the set,
obtained by summing the values and dividing by the number of values. Say, the
events x 1 , x 2 , x 3 ...

x N are occurring with frequencies n 1 , n 2 , n 3 ... n N respectively.

Therefore the average of occurrence of the events is given by

x 1 n 1 + x 2 n 2 + ....... + x N n N x 1 n 1 + x 2 n 2 + ....... + x N n N

Mean = μ = =

n 1 + n 2 + ....... + n N N

n 1
N +

n 2
N +

n N
N =

= x 1 x 2 ....... + x N x 1 p ( x 1 ) + x 2 p ( x 2 ) + ....... + x N

N

∴ μ = x i p ( x i )

i = 1

(A.3)

The variance of a data set is the arithmetic average of the squared differences
between the values and the mean. Again, when we summarize a data set in a
frequency distribution, we are approximating the data set by 'rounding' each value