Download chapter 6 continuous probability distributions

Continus Probability
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Continuous Probability
Distributions
• Uniform Probability Distribution
• Normal Probability Distribution
• Exponential Probability Distribution
(Optional)
f (x)Normal
Exponential
f (x)
f (x)Uniform
x
x
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
x
Continuous Probability Distributions
n
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.
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Data Types
Data
Numerical
(Quantitative)
Discrete
Continuous
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Categorical
(Qualitative)
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 Inter-arrival 0, 1.3, 2.78, ...
arrivals
time
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Continuous Probability
Distribution Models
In this
Chapter
Uniform
Continuous
Probability
Distribution
Normal
Exponential
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Other
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)
Uniform
x1 x 2
f (x)
x
Normal
x1 x2
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
x
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.
f(X)
Mean
Median
Business Statistics
Mode(BUSA 3101).
Dr.Lari H. Arjomand
X
Normal Probability Distribution
It has been used in a wide variety of applications:
Heights
of people
Scientific
measurements
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Normal Probability Distribution
• Normal Probability Density Function
1
 ( x   )2 /2 2
f (x) 
e
 2
where:
 = mean
 = standard deviation
 = 3.14159
e = 2.71828
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Normal Probability Distribution
Characteristics
1- The distribution is symmetric; its skewness
measure is zero.
x
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Normal Probability Distribution
Characteristics
2- The entire family of normal probability
distributions is defined by its mean  and its
standard deviation  .
Standard Deviation 
Mean 
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
x
Normal Probability Distribution
Characteristics
3- The highest point on the normal curve is at the
mean, which is also the median and mode.
x
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Normal Probability Distribution
Characteristics
4- The mean can be any numerical value: negative,
zero, or positive.
x
-10
0
20
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Normal Probability Distribution
Characteristics
5- The standard deviation determines the width of the
curve: larger values result in wider, flatter curves.
 = 15
 = 25
x
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Normal Probability Distribution
Characteristics
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).
.5
.5
x
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Normal Probability Distribution
Characteristics #7
68.26% of values of a normal random variable
are within +/- 1 standard deviation of its mean.
95.44% of values of a normal random variable
are within +/- 2 standard deviations of its mean.
99.72% of values of a normal random variable
are within +/- 3 standard deviations of its mean.
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
Normal Probability Distribution
Characteristics #7
99.72%
95.44%
68.26%
 – 3
 – 1
 – 2

 + 3
 + 1
 + 2
Business Statistics (BUSA 3101).
Dr.Lari H. Arjomand
x