Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continus Probability

Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

Continus Probability


Continuous Probability Distributions

• Uniform Probability Distribution• Normal Probability Distribution• Exponential Probability Distribution

(Optional)

f (x)f (x)

x x

Uniform

x

f (x)Normal

xx

f (x)f (x)Exponential


Continuous Probability Distributions A continuous random variable can assume any

value in an interval on the real line or in a collection of intervals.

It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval.


Data TypesData Types

Data

Numerical(Quantitative)

Categorical(Qualitative)

Discrete Continuous


Continuous Random Variable Examples

Continuous Random Variable Examples

Experiment Random Variable

Possible Values

Weigh 100 people Weight 45.1, 78, ...

Measure part life Hours 900, 875.9, ...

Ask food spending Spending 54.12, 42, ...

Measure time between arrivals

Inter-arrival time

0, 1.3, 2.78, ...


Continuous Probability Distribution Models

Continuous Probability Distribution Models

ContinuousProbabilityDistribution

Uniform Normal Exponential Other

In this

Ch

ap

ter


Continuous Probability Distributions

The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2.

f (x)f (x)

x x

Uniform

x1 x1 x2 x2

x

f (x) Normal

x1 x1 x2 x2


Normal Probability Distribution• The normal probability distribution is the

most important distribution for describing a continuous random variable.

• It is widely used in statistical inference.

X

f(X)

Mean Median Mode


Heightsof peopleHeights

of people

Normal Probability Distribution It has been used in a wide variety of

applications:

Scientific measurements

Scientific measurements


Normal Probability Distribution• Normal Probability Density Function

2 2( ) / 21( )

2xf x e

= mean = standard deviation = 3.14159e = 2.71828

where:


1- The distribution is symmetric; its skewness measure is zero. 1- The distribution is symmetric; its skewness measure is zero.

Normal Probability Distribution

Characteristics

x


2- The entire family of normal probability distributions is defined by its mean m and its standard deviation s .

2- The entire family of normal probability distributions is defined by its mean m and its standard deviation s .


Characteristics

Standard Deviation s

Mean mx


3- The highest point on the normal curve is at the mean, which is also the median and mode.3- The highest point on the normal curve is at the mean, which is also the median and mode.


Characteristics

x



Characteristics

-10 0 20

4- The mean can be any numerical value: negative, zero, or positive.4- The mean can be any numerical value: negative, zero, or positive.

x



Characteristics

s = 15

s = 25

5- The standard deviation determines the width of thecurve: larger values result in wider, flatter curves.5- The standard deviation determines the width of thecurve: larger values result in wider, flatter curves.

x


6- Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right).

6- Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right).


Characteristics

.5 .5

x



Characteristics #7

of values of a normal random variable are within of its mean.

of values of a normal random variable are within of its mean.68.26%68.26%

+/- 1 standard deviation+/- 1 standard deviation



+/- 2 standard deviations+/- 2 standard deviations



+/- 3 standard deviations+/- 3 standard deviations



Characteristics #7

xm – 3s m – 1s

m – 2sm + 1s

m + 2sm + 3sm

68.26%

95.44%99.72%

Documents

Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continus Probability