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U7 Continuous Random Variables
1
Module U7
Continuous Random Variables
Primary Author:
Email Address:
Last Update:
Prerequisite Competencies:
Module Objectives:
James D. McCalley, Iowa State University
[email protected]
7/12/02
Discrete Random Variables, Module U6
1. To distinguish continuous random variables from discrete
random variables.
2. To explain the relationship between a probability density
function and a probability mass function.
3. To obtain cumulative distribution functions from probability
density functions and vice-versa.
4. To use probability density functions and cumulative distribution
functions to obtain probabilities.
Overview
A continuous random variable (RV), like a discrete RV, is a real
numerically valued function of the experimental outcomes that is
defined over a sample space. It is the nature of the sample space that
determines which type of RV it is; RVs with countable sample spaces
are discrete; RVs with non-countable sample spaces are continuous.
U7.1
Probability Density Functions
In the discrete case, we used the probability mass function (PMF),
f X  x  to characterize the relation between the probability of the RV X
taking on the value x. This probability was given as an explicit non-zero
numerical value.
However, if we were to use a continuous RV, then we allow that our RV
X may take on any value x min  x  x max . Since there are an infinite
number of possible values that X may take, the probability that X is
exactly equal to a particular value, say 2.43985, is zero.
f X  x   lim
x  0
Px  X  x  x 
x
(U7.1)
U7 Continuous Random Variables
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Probability density functions can take a variety of shapes. To illustrate,
let’s look at two fairly simple PDFs, the uniform PDF and the triangular
PDF.
0.02,
f V v   
 0,
475  V  525
otherwise
(U7.2)
Figure U7.1 Uniform PDF for Instrument Measurement Error Example.
U7 Continuous Random Variables
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 0.16 d  0.8, 5  d  7.5

f D d    0.16 d  1.6, 7.5  d  10
 0,
otherwise

(U7.3)
Figure U7.2 Triangular PDF for Load Level Sample
One may observe from the above two examples that
1. f X  x   0 ( probability per unit RV, must be non-negative)
2.



U7.2
f X  x   1 (total probability, must be 1)
Evaluation of Probabilities Using PDFs
b
Pa  X  b   f X  x dx
a
(U7.7)
U7 Continuous Random Variables
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Recall the uniform distribution of Fig U7.1.
1. Between 475 and 525:
P475  V  525  
525
525
475 f V v dv  475 0.02 dv  1.0
2. Between 475 and 505:1:
P475  V  505  
505
505
 f V v dv  475 0.02 dv  0.6
475
3. Between 450 and 505:
P450  V  505 
505
505
 f vdv   0.02dv  0.6  P475  V  505
V
450
475
4. Less than 505:
P   V  505  
505
505
 f V v dv  475 0.02 dv  0.6
5. Greater than 505:
P505  V   

525
505 f V v dv  505 0.02 dv  0.4
All of the above calculations may be interpreted as areas under the curve
of Figure (U7.1)
U7.3
Cumulative Distribution Functions
P  X  b  P X  b
This probability has been given a specific name, the cumulative
probability function (CDF), denoted by F X  x  and given by
FX  x   P X  x 
x
 f X  d
(U7.9)
It is analogous to the CDF defined for the discrete case in that it gives
P X  x .
U7 Continuous Random Variables
Example IM4: The uniform pdf case.
0,
  v  475


FV v    0.02 v  9.5, 475  v  525

1.0,
525  v

Figure U7.3 CDF for Instrument Measurement Error Example
5
U7 Continuous Random Variables
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Example LL4: Triangular pdf case.
 0,

2
 0.08 d  0.8d  2,
FD  d   
0.5  0.08 d 2  1.6d  7.5,

1.0,

  d  5
5  d  7. 5
7.5  d  10
10  d
Figure U7.4 CDF for Load Level Example
Important properties of all continuous CDFs.
1. Its lower bound is zero, FX    0 , which must be the case
since FX    P X    0
2. Its upper bound is one, FX    1 , which must be the case
since FX   P X    1
3. All CDFs must be non-decreasing functions.
P X  x  P X  x  1  P X  x  1  FX  x  (U7.10)
4.
5.
6.
Pa  X  b  FX b  FX a 
d FX  x 
f X x 
dx