Page 137 - DMTH404_STATISTICS

Unit 9: Variance of a Random Variable and their Properties

9.2 Summary Notes

 The mean and variance of a random variable can be computed in a manner similar to the
computation of mean and variance of the variable of a frequency distribution.

 If X is a discrete random variable which can take values X , X , ..... X , with respective
1 2 n
probabilities as p(X ), p(X ), ...... p(X ), then its mean, also known as the Mathematical
1 2 n
Expectation or Expected Value of X, is given by:
n
E(X) = X p(X ) + X p(X ) + ...... + X p(X )   X p(X ) .
1 1 2 2 n n i i

i 1
The mean of a random variable or its probability distribution is often denoted by , i.e.,
E(X) = .
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
 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
  E X       X      , where X is a discrete random variable.
p X
i
i

i 1
9.3 Keywords
Random variable: If X is a discrete random variable which can take values X , X , ..... X , with
1 2 n
respective probabilities as p(X ), p(X ), ...... p(X ), then its mean, also known as the Mathematical
1 2 n
Expectation or Expected Value of X, is given by:
n
E(X) = X p(X ) + X p(X ) + ...... + X p(X )   X p(X ) .
1 1 2 2 n n i i
i 1

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.
Continuous random: 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

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