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### 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/
```
Associated Topics:
College Statistics

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