Biology Reference
In-Depth Information
binomial distribution introduced in the previous chapter provides
another example of a discrete distribution.
In contrast, if a random variable represents the weight of a newborn
baby, it can take on a continuous range of values. This is an example
of a continuous random variable, and the tool that replaces the table of
values with their corresponding probabilities is the probability density
function. A function f (x) is a probability distribution density function for
a random variable
x
provided that:
1
:
f
ð
x
Þ
0
;
Z
1
2
:
f
ð
x
Þ
dx
¼
1
;
1
3. For any numbers a<b, the probability that
x
is between a and b is
Þ¼
R
b
a
calculated as P
ð
a<
x
<b
f
ð
x
Þ
dx.
The most common and important probability density function is the
normal or Gaussian distribution. A special case is the standard normal
density function defined by the expression
0.50
1
2
e
x
2
f
ð
x
Þ¼
p
:
(4-1)
0.40
p
0.30
The graph of this function is depicted in Figure 4-1(A).
0.20
0.10
The mean and variance of a random variable are two numerical
characteristics associated with its distribution. The mean could be
thought of as the average we would expect after doing many trials. The
variance measures the spread of the data around the mean value. The
mean is typically represented by
A
0.00
−
4
−
3
−
2
−
1
0
1
2
3
4
0.50
0.40
2
. The standard
deviation, defined as the square root of the variance, is denoted by
m
and the variance by
s
0.30
s
.
0.20
The following heuristic explanation outlines a fundamental principle:
A normal distribution occurs when multiple independent random
choices are made, each of them attempting to achieve a certain fixed
average value, but each is vulnerable to errors that are symmetrical in
both directions around the mean.
0.10
B
0.00
−
6
−
4
−
2
0
2
4
6
8
10
FIGURE 4-1.
Comparison of normal density functions with
differing parameters. Panel A: Standard normal
density; that is, normal density with mean
m ¼
Several examples will clarify this point. Thousands of bats exit their
cave about 2 hours after sunset. Because inside the cave there is
no indication of when exactly sunset occurs, each bat relies on its
biological clock to estimate the exit time. As we shall see in a later
chapter, some biological clocks run fast and some slow, with larger
errors less likely to occur than smaller ones. Experiments have shown
that the number of bats exiting a cave per minute follows quite
precisely a bell-shaped curve with a mean of about 2 hours after sunset.
Similarly, if we place microscopic particles in a glass of water, they get
hit by water molecules and travel various distances, depending on the
0 and standard deviation
s ¼
1; panel B:
normal densities with
m ¼
2 and
s
1
¼
3 (gray
line),
s
2
¼
2 (black dotted line), and
s
3
¼
1 (solid
black line).
