Ch4.3 Normal Distribution
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Transcript Ch4.3 Normal Distribution
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
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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%
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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