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##

Variance 11/10

*E*[*X*] does not say anything about the
the spread of the values.
This is measured by

which can also be written in the **computational formula**:

The unit in which this is measured
is not coherent with that of *X*, we very often
use the standard deviation

For the Bernouill(p): *X*^{2}=*X*

*var*(*X*)=*E*(*X*^{2})-(*E*(*X*))^{2}=*p*-*p*^{2}=*p*(1-*p*)=*pq*

Property 1:

For two independent random variables, X and Y:

*Var*(*X*+*Y*)=*Var*(*X*)+*Var*(*Y*)

This essential property allowed us to
compute the variance of a binomial *S*_{n}, because we can write
a binomial as the sum of n independent Bernouilli(p)
random variables *X*_{i} so that:

Example(which I `did' in class!):

What is the variance of the geometric?
Using the computational formula, we compute first *E*(*X*^{2}):

From the sums of independent variables theorem,
and the fact that a Negative Binomial *Y*_{r} can be written
as the sum of *r* independent Geometrics, we have:

I also showed in class, that the variance of the Poisson()
random variable is ,
a fact that helps
recognize a Poisson random variable.

** Next:** Conditional Expectation
** Up:** Expectation and Variance 11/6
** Previous:** Conditional Expectation 11/9
*Susan Holmes*

*1998-12-07*