Home >>STATISTICS, Section 1, discrete random variables 3

Variance of a discrete random variable Var(X)

By definition,

variance is a measure of the spread(dispersion) of a set of data points around their mean value.

Mathematically, the variance is the expectation( mean) of the averageof the squared deviations from the mean.

where,

x_r is any value of the random variable x
μ is the mean value of x
n is the number of values of x
Σ means the sum of all values in the brackets

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Variance and Expected Values

Variance can also be written in terms of the expectation operator E( ) :

This can be expanded and consolidated to another form:

Substituting for E[X] from μ = E[X] :

or

So the mean μ , the variance Var(X) and the standard deviation σ (sigma) are all related to one another.

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Variance and Linearity

In a similar way to the Expectation Operator for E[aX + b] , the Variance Operator Var( ) behaves in a similar, though not identical, fashion.

However,                

This can be derived from first principles.

Using,  

(from above)

and replacing X with aX+b,

since,                          

then,                    

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