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MTH 246
(Normal Probability Distribution)
The most important probability distribution in the entire …eld of statistics is the normal distribution.
Originally, the mathematical equation of the normal distribution was developed by Abraham DeMoivre in
1733. However, it is often referred to as the Gaussian distribution in honor of Karl Friedrich Gauss (17771855). The normal distribution is important both because it seems to provide an adequate model for various
observed measurements and, as we will see in the subsequent chapters, because it provides an accurate approximation to a wide variety of probability distributions.
The graph of the normal distribution is a bell-shaped curve and is called the normal curve or Gaussian
curve.
A continuous random variable X that has the bell-shaped normal distribution is called a normal random
variable or Gaussian random variable. The probability distribution of the normal random variable is a function of the form as shown in the following de…nition. It depends upon the two parameters and > 0 that
can be shown to be the mean and standard deviation, respectively, of the normal random variable.
De…nition 1. A random variable X is said to have a normal probability distribution (or Gaussian
distribution) with parameters and if its probability density function is given by
"
#
2
1 x
1
f (x) = p exp
2
2
for
1<
When
That is,
< 1,
= 0 and
> 0, and
1 < x < 1.
= 1, the normal probability distribution is called the standard normal distribution.
De…nition 2. The probability distribution of a normal random variable X with mean
= 0 and
standard deviation = 1 is called a standard normal distribution. That is, the probability distribution of
the random variable X is given by
1
f (x) = p exp
2
Example 7 (on page 613 ).
deviation = 16, then
P (300
1 2
x
2
for
1<x<1
If X is a normal random variable with mean
X < 1) =
Z1
300
1
p exp
16 2
"
1
2
x
268
16
By read the probability table, this probability can be shown to be 0:02275.
1
2
#
dx:
= 268 and standard