The expected value is a weighted average of the values of a random variable may assume. The weights are the probabilities.

**Expected
value and Variance**

The expected value is a
weighted average of the values of a random variable may assume. The weights are
the probabilities.

Let *X* be a
discrete random variable with probability mass function (p.m.f.) *p*(*x*)
. Then, its expected value is defined by

If *X* is a
continuous random variable and *f*(*x*) is the value of its
probability density function at *x*, the expected value of *X* is

**Note**

In (1),** ***E*(*X*) is defined
to be the indicated series provided that the series is** **absolutely convergent;
otherwise, we say that the mean does not exist.

In (1),** ***E*(*X*) is an
“average” of the values that the random variable takes on,** **where each value is
weighted by the probability that the random variable is equal to that value.
Values that are more probable receive more weight.

In (2),** ***E*(*X*) is defined
to be the indicated integral if the integral exists;** **otherwise, we say that
the mean does not exist.

In (2),** ***E*(*X*) is an
“average” of the values that the random variable takes on,** **where each value x is
multiplied by the approximate probability that *X* equals the value *x*,
namely *f* * _{X}* (

The variance is a
weighted average of the squared deviations of a random variable from its mean.
The weights are the probabilities. The mean of a random variable *X*,
defined in (1) and (2), was a measure of central location of the density of *X*.
The variance of a random variable *X* will be a measure of the spread or
dispersion of the density of *X* or simply the variability in the values
of a random variable.

The variance of *X*
is defined by

if *X* is
discrete random variable with probability mass function *p*(*x*).

if *X* is
continuous random variable with probability density function *f* * _{X}*
(

Expected value of [*X*
–*E*(*X*)]^{2} is called the variance of the random variable.

**Note**

·
In the following examples, variance will be found using
definition 6.10.

·
The variances are defined only if the series in (3) is
convergent or if the integrals in (4) exist.

·
If X is a random variable, the standard deviation of X (S.D(X)),
denoted by σ_{X} , is defined as

·
The variance of X , denoted by σ_{X}^{2}
or Var(X) or V(X)

Tags : Definition, Formulas | Mathematical Expectation , 12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation

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12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation : Expected value and Variance | Definition, Formulas | Mathematical Expectation

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