Biology Reference
In-Depth Information
force of the impact. The average travel distance depends on the
temperature (energy) of the water, while the distribution of the
distances around this average is approximately normal. In other
words, each bat's time estimate of 2 hours after sunset, or each
particle's travel distance, is one outcome of an experiment described
by a random variable with a normal density function. The general
analytical expression of the normal (Gaussian) density is given by the
function
1
2
x
m
s
Þ
2
e
ð
ð
Þ¼
p
;
f
x
(4-2)
p
s
where x can take any value in the interval (
1
,
1
). When we say
that a random variable
x
has a normal distribution with parameters
m
and
s
, this means the random quantity represented by
x
has a density
function such as in Eq. (4-2). The values
define uniquely the
shape of the curve and correspond to the mean and the standard
deviation of the normal distribution. The mean value
m
and
s
,
determines the position of the maximum for the bell-shaped graph of
the normal density function from Eq. (4-1). The standard
deviation
m
,
1
<
m
<
1
0, determines how sharp the peak is near the maximum.
Figure 4-1(B) illustrates how the density graphs for the function in
Eq. (4-2) change with the change of the parameters
s
,
s >
m
and
s
.
When
1, the density function from Eq. (4-2) takes
the form presented by Eq. (4-1). That is, for
m ¼
0 and
s ¼
m ¼
0 and
s ¼
1, we obtain
the standard normal distribution.
If the outcomes of two or more experiments do not influence one
another, we refer to them as being independent. Similarly, we say that
the random variables associated with these experiments are also
independent. Consider three random variables describing the
following quantities: (1) The time after sunset when a bat exits its
cave located in Virginia; (2) the height of a corn plant in a cornfield in
Mexico; and (3) the acidity of the soil on which the same plant
grows. Because it appears unlikely that the characteristics of the
cornfield in Mexico can affect the behavior of Virginia bats, the bat exit
time can be considered independent from the corn height and also
independent from the acidity of the cornfield. On the other hand,
theacidityofthesoilandtheheightofthecornplantscannotbe
considered independent because the acidity is expected to
affect growth.
If we combine two random variables with the same distribution,
the result will not necessarily have that same distribution. For example,
if we flip a coin twice and let
1
be the number of heads (H) on
the first flip and
2
be the number of heads on the second flip, then the
probabilities that
1
takes values 1 or 0, respectively, are P
ð
1
¼
1
Þ¼
1
=
2
ð
1
¼
Þ¼
=
2
will have exactly
and P
0
1
2. The random variable
