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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): X2=X

var(X)=E(X2)-(E(X))2=p-p2=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 Sn, because we can write a binomial as the sum of n independent Bernouilli(p) random variables Xi so that:

Example(which I `did' in class!):
What is the variance of the geometric? Using the computational formula, we compute first E(X2):

From the sums of independent variables theorem, and the fact that a Negative Binomial Yr 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