Download normal distribution

NORMAL DISTRIBUTION
• The probability density function
– f(x), area and the effect of changing mean and variances
• Standard normal distribution
– Relationship between x and z
– Table, z values, area and probability
– Given z find area, use of table and Excel
– Given area find z, notation zA, and use of table and Excel
• Given x, find probability
• Given probability, find x
• Normal approximation to binomial and continuity correction
1
NORMAL DISTRIBUTION
THE PROBABILITY DENSITY FUNCTION
• If a random variable X with mean  and standard deviation
 is normally distributed, then its probability density function
is given by
 1  1/ 2  x   /  2
f x   
e
  2 
2
NORMAL DISTRIBUTION
THE PROBABILITY DENSITY FUNCTION
f (x )
Mean, =50
SD, =10
0.0500
f (x )
0.0400
Area between
the vertical lines
= P (40x  60)
0.0300
Area under
the curve
= 1.00
0.0200
0.0100
0.0000
0
20
40
60
x - VALUES
80
100
3
NORMAL DISTRIBUTION
EFFECT OF CHANGING STANDARD DEVIATION
SD,=10
0.0500
SD,=15
f(x)
0.0400
SD,=20
0.0300
0.0200
Mean,= 100
0.0100
0.0000
0
50
100
150
200
4
x-values
NORMAL DISTRIBUTION
EFFECT OF CHANGING MEAN
0.0500
f(x)
0.0400
0.0300
0.0200
SD,10
0.0100
0.0000
50
100
150
x -values
200
250
5
STANDARD NORMAL DISTRIBUTION
0.4500
Mean=0
SD=1
0.4000
0.3500
f (x )
0.3000
0.2500
0.2000
0.1500
Area under
the curve = 1
0.1000
0.0500
0.0000
-6
-4
-2
0
z -VALUES
2
4
6
6