Prove that in general the variance of X can be computed as f

Prove that, in general, the variance of X can be computed as follows sigma_X^2 = F{X^2} - [E{X}]^2 For convenience, define the notation mu_X = E{X}.

Solution

The expected value of a random variable gives a crude measure of the “center of loca-
tion” of the distribution of that random variable. For instance, if the distribution is symmet-
ric about a value µ then the expected value equals µ.
To refine the picture of a distribution distributed about its “center of location” we need
some measure of spread (or concentration) around that value. The simplest measure to cal-
culate for many distributions is the variance. There is an enormous body of probability •variance literature that deals with approximations to distributions, and bounds for probabilities and
expectations, expressible in terms of expected values and variances.
<4.1> Definition. The variance of a random variable X with expected value EX = µX is
defined as var(X) = E¡
(X µX )2
¢
. The covariance between random variables Y and •covariance Z, with expected values µY and µZ , is defined as cov(Y, Z) = E((Y µY )(Z µZ )). The
correlation between Y and Z is defined as •correlation
corr(Y, Z) = cov(Y, Z)
q
var(Y )var(Z)
The square root of the variance of a random variable is called its standard deviation. ¤ •standard deviation
As with expectations, variances and covariances can also be calculated conditionally on
various pieces of information.
Try not to confuse properties of expected values with properties of variances. For ex-
ample, if a given piece of “information” implies that a random variable X must take the con-
stant value C then E(X | information) = C, but var(X | information) = 0. More generally, if
the information implies that X must equal a constant then cov(X, Y ) = 0 for every random
variable Y . (You should check these assertions; they follow directly from the Definition.)
Notice that cov(X, X) = var(X). Results about covariances contain results about vari-
ances as special cases.
A few facts about variances and covariances
Write µY for EY , and so on, as above.
(i) cov(Y, Z) = E(Y Z) (EY )(EZ) and, in particular, var(X) = E(X2) (EX)2:
cov(Y, Z) = E(Y Z µY Z µZ Y + µYµZ )
= E(Y Z) µYEZ µZEY + µYµZ
= E(Y Z) µYµZ