Download Continuous Probability Distribution

Continuous Probability
Distribution
• Introduction:
•
A continuous random variable has
infinity many values, and those values
are
often
associated
with
measurements on a continuous scale
with no gaps.
Continuous Probability
Distribution
• Continuous uniform distribution:
• It is the simplest continuous
distribution above all the statistics
continuous distribution.
• This distribution is characterized by a
density functions flat and thus the
probability is uniform in a closed
interval [A,B].
Continuous Probability
Distribution
• The density function of a uniform
continuous distribution X on interval
[A,B] is:
 1
f ( X ; A, B)   B  A

0

, A  X  B

, otherwise 
Continuous Probability
Distribution
• The density function forms a rectangle
with base [B-A] and constant height
1
BA
so that the uniform distribution is often
called the rectangular distribution.
Continuous Probability
Distribution
• Uniform distribution
Continuous Probability
Distribution
Uses of uniform distribution:
1. In risk analysis.
2. The position of a particular air
molecule in a room.
3. The point on a car tire where the next
puncture will occur.
4. The length of time that some one
needs to wait for a service.
Continuous Probability
Distribution
•
Mean and variance of a uniform
distribution
b
b
1
mean(  )  E ( X )   xf ( x)dx   x
dx
ba
a
a
1 1 2 
1
ba
2
2

x  
(b  a ) 

b  a  2  a 2(b  a )
2
b
Continuous Probability
Distribution
•
Variance
var iance( 2 )  V ( x)  E ( X 2 )  ( E ( X )) 2
b
E( X ) 
2

a
1
1
b3  a 3
3 b
X .
dx 
( X ]a ) 
(b  a )
3(b  a)
3(b  a)
2
(b  a )(b 2  ab  a 2

3(b  a )
b 2  ab  a 2 b 2  2ab  a 2 b 2  2ab  a 2 b  a 
V (X ) 



3
4
12
12
2
Continuous Probability
Distribution
•
•
Example 1:
The continuous random variable X
has a probability distribution function
(f(x)) as the figure bellow
Continuous Probability
Distribution
•
Example 1:
Continuous Probability
Distribution
• Find:
1. The value of k.
2. P(2.1  X  3.4)
3. E(X)
Continuous Probability
Distribution
•
•
Solution:
The area under the curve must be
equal 1. Then
1
k  1  0
4
k 1  4
k  5
1
1.3
P(2.1  X  3.4)  (3.4  2.1) 
4
4
1 5
E( X ) 
3
2
Continuous Probability
Distribution
•
•
•
Example 2:
The current in (mA) measured in a piece of
copper wire is known to follow a uniform
distribution over the interval [0,25]. Write
down the formula for probability density
function f(X) of random variable X
representing the current. Calculate mean
and variance of distribution.
Solution:
Continuous Probability
Distribution
Solution:
1
1
 1



f ( x)  b  a
25 - 0 25

otherwise
0

0  X  25


b  a 25
E( X ) 

 12.5
2
2
(b  a) 2 25 2
V (X ) 

 52.08
12
12
m(A)
m(A 2 )
Continuous Probability
Distribution
•
•
Example 3:
Suppose that a large conference room at a
certain company can be reserved for no
more than 4 hours. Both long and short
conference occurs quite often. In fact it can
be assumed that the length X of a
conference has a uniform distribution on
the interval [0,4]
1. What is the probability density function?
2. What is the probability that any given
conference at least 3 hours?
3. Calculate mean?
Continuous Probability
Distribution
•
Solution:
1
1. f(X)   4

0

0  X  4

otherwise 
4
1
1
2. P(X  3)   dx 
4
4
3
4
1
11 2 4 1
3. E(X)   xdx 
x ]0   16  2
4
42
8
0