Chapter 7 – The Normal Probability Distribution
The normal distribution is a special type of bellshaped curve.
Defn
:
A random variable X is said to be normally distributed
or to have a normal distribution
if its
distribution has the shape of a normal curve:
(
29
(
29
2
2
2
2
1
σ
μ
σ
π


=
x
e
x
f
, for 
∞
< x <
∞
, 
∞
<
μ
<
∞
, and
σ
> 0.
Here
μ
and
σ
are the parameters of the distribution;
μ
= the mean of the random variable X (or of the
probability distribution); and
σ
= the standard deviation of X.
The two mathematical constants
appearing in the formula,
π
2245
3.141592654, and e
2245
2.718281828.
The normal distribution is not just a single distribution, but rather a family of distributions; each
member of the family is characterized by a particular pair of values of
μ
and
σ
.
The mean,
μ
, tells us
the location of the center of the bellshaped curve.
The standard deviation,
σ
, tells us how wide the
bellshaped curve is; smaller values of
σ
denote narrower, more sharply peaked bellshaped curves;
larger values of
σ
denote wider, less sharply peaked bellshaped curves.
In particular, the points given
by
μ

σ
and
μ
+
σ
are the inflection points of the curve.
Examples of Normal Distributions:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Characteristics of normal distributions
:
1)
The bell is symmetric around the mean value
μ
.
2)
The standard deviation,
σ
, describes the width of the bell.
3)
The distribution is unimodal (one peak), and the mean, median and mode are all equal.
4)
The curve is continuous; i.e., there are no gaps.
5)
For any two numbers, a and b, the probability that the random variable X will be found to have a
value between a and b is denoted by P(a
≤
X
≤
This is the end of the preview.
Sign up
to
access the rest of the document.