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