Elementary Statistics:
A Review
Expected Values
); Specifically,
the mean of
X,
denoted
Px,
is
defined
by
..
Px
=
E(X)
=
P,X,
+
p,X,
+ ...
+
P"X ..
=
LP,X,
;=1
Expected Values
);. Random variables are often described in terms of their
means and variances, which in turn are defined in terms of
the
expectations
operator E.
);. Assume that
X" X
2
,
••• ,
X
N
represent the
N
possible
outcomes associated with the random variable
X.
l>
Then the
mean,
or
expected value,
of
X
is a weighted
average of the possible outcomes, where the probabilities
of the outcomes serve as the appropriate weights.
where
Pi
is the probability
tbatX,
occurs,
!:Pi
=
I, and
E( )
is the expectations
operator.
Expected Values: Variance
l>
The
variance
of a random variable provides a measure of
the spread, or dispersion, around the mean.
);.It
is denoted
a~
and it is defined as
..
Var(X)
=
0;
=
I>,
[X, E(X)j'
r=)
);. Thus, the variance is a weighted average of the squares of
the deviations of outcomes on
X
from its expected value,
with the corresponding
probabilities
of each outcome
occurring serving as weights.
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 Fall '10
 Sapp
 Variance, Probability theory, expectations operator

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