Download Normal distribution

Normal distribution (2)
When it is not the standard
normal distribution
The Normal Distribution
WRITTEN :
X ~ N ( , )
2
… which means the
continuous random variable X is
normally distributed with mean 
and variance 2 (standard deviation )
The Standard Normal Distribution
• The random variable is called Z
• Z is called the standard normal distribution
– its mean  is 0
– standard deviation  is 1
• The distribution function is denoted by 
• Area under the curve = probability
Z ~ N (0,1)
(Z)
The Standard Normal Distribution
Z ~ N (0,1)
The probabilities are
given by the area under
the curve
(-1.6) = P(Z<-1.6)
=0.0548
By symmetry:
(1.6) =1 - (-1.6)
P(Z<-1.6) = 1 - P(Z<1.6)
Probability above 75?
Probability student scores higher than 75?
0.08
0.07
Density
0.06
0.05
P(X > 75)
0.04
0.03
0.02
0.01
0.00
55
60
65
70
75
Grades
2
X ~ N (70,5 )
80
85
The Normal Distribution
X ~ N ( , )
2
The Standard Normal Distribution
Z ~ N (0,1)
•Tables are for standardised Z
•May want to find other solutions (given  and 2)
•The normal distributions must be ‘standardised’
•However, GDCs can handle either
Standardising
X ~ N ( , )
2
Use the
transformation
Z
Z ~ N (0,1)
X 

…. then, use probability table for Z
Probability above 75?
Probability student scores higher than 75?
0.08
0.07
2
X ~ N (70,5 )
Density
0.06
0.05
P(X > 75)
0.04
P(X>75)
0.03
0.02
0.01
0.00
55
60
65
70
Grades
Z
X 

1 - P(X<75)
75
80
85
= 1 - P(X<75)
75  70
Z
5
1 - P(Z<1)
= 1 - 0.8413
= 0.1587
Probability between 65 and 70?
2
X ~ N (70,5 )
0.08
0.07
Density
0.06
0.05
P(65 < X < 70)
0.04
0.03
0.02
0.01
0.00
55
60
65
70
75
80
85
Grades
P(65<X<70) = P(X<70) - P(X<65)
Probability between 65 and 70?
2
X ~ N (70,5 )
0.08
0.07
Density
0.06
0.05
P(65 < X < 70)
0.04
0.03
0.02
0.01
0.00
55
60
65
70
75
80
85
Grades
P(65<X<70) = P(X<70) - P(X<65)
Z
X 

P(-1<Z<0)
70  70
65  70
Z
Z
5
5
P(Z<0) - P(Z<-1)
P(Z<0) - [1- P(Z<1)]
0.5
-
[1 - 0.8413] = 0.3413
Probability between 65 and 70?
Why not GDC?
0.08
0.07
Density
0.06
0.05
normalcdf(lower bound, upper bound,
mean (), standard deviation ())
P(65 < X < 70)
0.04
0.03
0.02
0.01
0.00
55
60
65
70
75
80
85
Grades
2
X ~ N (70,5 ) P(65<X<70)
2nd
distr 2 normalcdf(65, 70, 70, 5)
Z ~ N (0,1) P(-1<Z<0)
2nd
distr 2 normalcdf(-1, 0)
(if you close bracket it assumes ‘Z’)
Probability above 75?
Probability student scores higher than 75?
0.08
0.07
2
X ~ N (70,5 )
0.06
Density
0.05
P(X > 75)
0.04
0.03
0.02
0.01
0.00
55
60
65
70
75
80
85
Grades
normalcdf(lower bound, upper bound,
mean (), standard deviation ())
No upper bound!!!!
2nd
distr 2 normalcdf(75, E99, 70, 5)
Very very big
Probability below 65?
0.08
0.07
2
0.06
X ~ N (70,5 )
Density
0.05
0.04
0.03
0.02
P(X < 65)
0.01
0.00
55
65
75
85
Grades
normalcdf(lower bound, upper bound,
mean (), standard deviation ())
No lower bound!!!!
2nd
distr 2 normalcdf(-E99, 65, 70, 5)
Very very small