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