Section 1 : Discrete Random Variables(3)
 

[variance Var(X) ][variance & expected values]

[variance and linearity]

 

 

 

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.

definition of variance

Where,

xr 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( ) :

variance in terms of the expectation operator

This can be expanded and consolidated to another form:

variance and expected value

substituting for E[X] from μ = E[X]

variance and expected value

or

variance and expecteed value

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

Var(X) = σ2

 

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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.

        E[aX + b] = aE(X) + b

however,                Var(aX + b) = a2 Var(X)

This can be derived from first principles.

using             variance and expecteed value     (from above)

and replacing X with aX+b,

variance linearity

since,                           Var(X) = E[X2] - E[X]2

then,                    Var(aX + b) = a2 Var(X)

 

 

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