Variance and Expected Value

Date: 09/12/2004 at 19:57:58
From: Frank
Subject: Variance is equal to the expected value 

How can you prove that the variance is equal to the expected value of
the square of the random variable minus the mean squared?

Date: 09/13/2004 at 09:32:23
From: Doctor Luis
Subject: Re: Variance is equal to the expected value 

Hi Frank,

For a random variable X, the mean mu is the expected value E[X], and 
the variance Var(X) is the expected value of (X - mu)^2. So,

     mu  = E[X]

  Var(X) = E[(X - mu)^2]
         = E[X^2 - 2mu*X + mu^2]
         = E[X^2] - E[2mu*X] + E[mu^2]
         = E[X^2] - 2mu*E[X] + mu^2
         = E[X^2] - 2mu*mu + mu^2
         = E[X^2] - 2mu^2 + mu^2
         = E[X^2] - mu^2

where I've used several properties for the expected value, namely 
E[U + V] = E[U] + E[V], E[aU] = a*E[U], E[b] = b, for random variables 
U,V and constants a,b.

Note that Var(X) can also be written as E[X^2] - E[X]^2, which makes 
it easy to remember.

I hope this helped!  Please write back if you have further questions.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/