Ch4.3 Normal Distribution
A continuous RV X is said to have a normal distribution with
 and  , where       and
parameters
0   , if the pdf of X is
1
 ( x   )2 /(2 2 )
f ( x) 
e
 2
  x  
The normal distribution with parameter values
  0 and   1 is called a standard normal distribution.
The random variable is denoted by Z. The pdf is
1
z2 / 2
f ( z;0,1) 
e
z
2
The cdf is ( z )  P( Z  z )   f ( y;0,1)dy

Ch4.3
Ch4.3
Standard Normal Cumulative Areas
Shaded area =  (z )
Standard
normal
curve
z will denote the value on the measurement axis for
which the area under the z curve lies to the right of
Shaded area  P( Z  z )  
z
Ch4.3
Ch4.3
Nonstandard Normal Distributions
If X has a normal distribution with mean
deviation  , then
X 
Z 

has a standard normal distribution.
Ch4.3
and standard
Ch4.3
Normal Curve
Approximate percentage of area within given standard
deviations (empirical rule).
99.7%
95%
68%
Ch4.3
Ch4.3
Percentiles of an Arbitrary Normal Distribution
(100 p)th for

(100p)th percentile for



normal
standard normal 
Normal Approximation to the Binomial Distribution
Let X be a binomial rv based on n trials, each with
probability of success p. If the binomial probability
histogram is not too skewed, X may be approximated by
a normal distribution with   np and   npq .
 x  0.5  np 
P( X  x)   


npq


Ch4.3