Page 136 - DMTH404_STATISTICS

Statistics

Notes Remarks: The mean of a frequency distribution can be written as

f f f
X . 1  X . 2  ......  X . n , which is identical to the expression for expected value.
1 2 n
N N N
Variance

The concept of variance of a random variable or its probability distribution is also similar to the
concept of the variance of a frequency distribution.

The variance of a frequency distribution is given by

1 2 2 f 2
2
   i  f X  X    X  X  . i = Mean of  X  X  values.
i
i
i
N N
The expression for variance of a probability distribution with mean  can be written in a similar
way, as given below :
n
2
2
2
p X
  E X       X      , where X is a discrete random variable.
i
i

i 1
Remarks: If X is a continuous random variable with probability density function p(X), then

E X
     X.p(X)dX
2
2
2
  E X         X    .p(X)dX
9.1.1 Moments
The rth moment of a discrete random variable about its mean is defined as:
n r
r
 
  E X    X    p(X )
i
r
i

i 1
Similarly, the rth moment about any arbitrary value A, can be written as
n
r r
   E X A    X  A  p(X )
r i i

i 1
The expressions for the central and the raw moments, when X is a continuous random variable,
can be written as
r
r

  E X      X    .p(X)dX
r  
r
and    E X A      X A .p(X)dX  r respectively.
r

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