Post on 01-Jan-2016
Normal distribution
and intro to continuous probability density functions...
Discrete Distribution
Mean = np = 10 x 0.5 = 5
Symmetrical Binomial Distribution B(10, 0.5)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
r
Prob
P(X=r)
As a Histogram(Area of rectangle = probability)
Symmetrical Binomial Distribution B(10, 0.5)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
r
Prob
P(X=r)
Decrease interval size...Symmetrical Binomial Distribution
B(30, 0.5)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
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
Prob
P(X=r)
Decrease interval size more….
Binomial Distribution : B(100,0.5)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
r
Prob
P(X=r)
Almost a nice continuous curve
Continuous probabilitydensity 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
),(~ 2NXWRITTEN :
… which means the continuous random variable X is normally distributed with mean and variance 2
Examples of normalrandom variables
55 60 65 70 75 80 85
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Grades
Den
sityProbability student scores higher than 75?
P(X > 75)
Properties of Normal Distribution