Download Normal distribution

Normal distribution
and intro to continuous probability
density functions...
Discrete Distribution
Symmetrical Binomial Distribution
B(10, 0.5)
0.3
P(X=r)
0.25
Prob
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
r
Mean = np = 10 x 0.5 = 5
8
9
10
As a Histogram
(Area of rectangle = probability)
Symmetrical Binomial Distribution
B(10, 0.5)
0.3
P(X=r)
0.25
Prob
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
r
6
7
8
9
10
Decrease interval size...
Symmetrical Binomial Distribution
B(30, 0.5)
0.16
P(X=r)
0.14
0.12
Prob
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
r
Decrease interval size more….
0.09
Binomial Distribution :
B(100,0.5)
P(X=r)
0.08
0.07
0.05
Almost a nice
continuous curve
0.04
0.03
0.02
0.01
r
95
10
0
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
0
Prob
0.06
Continuous probability
density functions
• The curve describes probability of getting
any range of values, say P(X > 60),
P(X<30), P(20 < X < 50)
• Area under the curve = probability
• Area under whole curve = 1
• Probability of getting specific number is 0,
e.g. P(X=60) = 0
Characteristics of
normal distribution
• Symmetric, bell-shaped curve.
• Shape of curve depends on population
mean  and variance 2.
• Center of distribution is .
• Spread is determined by .
• Most values fall around the mean, but some
values are smaller and some are larger.
• Probabilities are from area under the curve
The Normal Distribution
WRITTEN :
X ~ N ( , )
2
… which means the
continuous random variable X is
normally distributed with mean 
and variance 2
Examples of normal
random variables
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
Grades
75
80
85
Properties of Normal Distribution