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Math:Probability:Distribution:Exponential: Can you describe Exponential distribution L . e^(-L . t)?

Feb 20th, 2005 02:35
Knud van Eeden,


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--- Knud van Eeden --- 28 December 2004 - 09:35 pm -------------------
Math:Probability:Distribution:Exponential: Can you describe 
Exponential distribution L . e^(-L . t)?
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The Exponential distribution is defined as
             (-L . t)
 f(t) = L . e
or
                        (-constant . x)
 y = f(x) = constant . e
---
---
e.g. choosing L equal to 1 gives:
          (-1 . x )    -x
 y = 1 . e          = e
e.g. choosing L equal to 3 gives:
          (-3 . x)
 y = 3 . e
---
---
It is a continuous distribution.
---
---
History:
Karl Pearson used the naming "negative exponential curve" the first
time in 1895.
---
---
Note:
The 'Exponential' distribution has got his name because of the
'exponential' function (=e^...) in the distribution function
above.
---
---
Note:
It is also called the Boltzmann distribution.
---
---
The Exponential distribution is mainly used to model the next
occurrence of an event in Poisson processes.
For example:
time:
-What is the waiting time until the next telephone call arrives?
length:
-What is the distance until the next flaw in a copper wire?
area:
-What is the distance until the next crack in a thin metal plate?
volume:
-What is the distance until the next occurrence of a raisin in a cake?
---
---
Note:
Scaling simultaneously in the vertical direction and the horizontal 
direction
You see that constant L, which happens to be the expectation (or the
mean or average) of the Poisson distribution, functions like a linear
scaling factor
1. so linearly scaling the vertical y values of this function
 y = L . (...) = constant . (...) = (linear scaling factor) . (...)
2. and also simultaneously as a scaling factor in the power of that e 
function
            (-L . x)
 y = ... . e
                 (-constant . x)
        = ... . e
                 ((linear scaling factor) . x)
        = ... . e
Now this basically scales graph on the independent variable axis.
---
Making this scaling factor larger squeezes the graph (with the same 
height)
together on the independent variable axis
(because you take an x value further away on the x axis (via this
larger scale factor), calculate the corresponding y value there, and 
put this
result back on this original x position. If you do this for all the x
of that graph it will effectively squeeze that graph together).
e.g.
 sin( x )
versus
 sin( 100 . x )
(this takes x values 100 times further away, calculates the
 corresponding y value there, and then puts this corresponding y value 
on the
 original x position. So this will press the graph together if you do
 this for all x, thus squeezing that graph).
---
Making this scaling factor smaller stretches the graph (with the same 
height)
out on the independent variable axis.
(because you take an x value closer (via this smaller scale factor),
calculate the corresponding y value there, and put this result back on
this original x position. If you do this for all the x of that graph it
will effectively stretch that graph out).
e.g.
 sin( x )
versus
 sin( 1/100 . x )
(this takes x values 100 times closer, calculates the
 corresponding y value there, and then puts this corresponding y value 
on the
 original x position. So this will press the graph together if you do
 this for all x, thus stretching that graph).
---
So if you let this constant L act simultaneously it will
1. simultaneously linearly scale the graph over the independent x axis,
2. and also linearly scale it with the same factor over the dependent y
    axis.
---
Thus if you make L, the constant, larger, it will simultaneously
press the graph together over the x axis and make the height larger.
Thus if you make L, the constant, smaller, it will simultaneously
stretch the graph out over the x axis and make the height smaller.
---
e.g.
Choosing L equals to 1 gives the negative exponential function:
          (-1 . x )    -x
 y = 1 . e          = e
Choosing L equals to 3 will
 1. press that graph 3 times together on the x axis
 2. also make that graph's height 3 times as large
          (-3 . x)
 y = 3 . e
---
---
Book: see also:
---
[book: Bronshtein / Semendyayev - a guide book to mathematics - ISBN 0-
387-91106-5 - p. 609 'Exponential distribution' (short formula)]
---
[book: Daintith, John / Nelson, R. David - the penguin dictionary of 
mathematics - ISBN 0-14-051119-9 - p. 142 'Gamma distribution' (The 
case a=1 gives the 'Exponential distribution' important in 'waiting 
time' problems (=the distribution of time from zero to the first 
occurrence of an event, and of the interval between future 
occurrences)]
---
[book: Hastings, Kevin J. - Probability and statistics - ISBN 0-201-
59278-9 - p. 133 'Exponential density' (good explanation, in depth 
discussion further)]
---
[book: Montgomery, Douglas, C. / Runger, George C. / Hubele, Norma, 
F. - engineering statistics - publisher: John Wiley - ISBN 0-471-17026-
7 - p. 99 'Exponential distribution' (with 3 good examples and an in 
depth explanation) / memorylessness]
---
[book: Spanier, Jerome / Oldham, Keith B. - an atlas of functions - 
ISBN 0-89116-573-8 - p. 260 'Exponential or Boltzmann distribution' 
(only a few short formulas)]
---
[book: Spiegel, Murray R. - Probability and statistics - McGraw-Hill 
(Schaum series) - p. 119 'The Exponential distribution' (only a very 
short formula)]
---
[book: Weisstein, Eric W. - CRC concise encyclopedia of mathematics - 
ISBN 0-8493-9640-9 - p. 595 'Exponential distribution' (1 
graph, 'memoryless random function', lot of formulas about this curve)]
---
[book: Zwillinger, Daniel - CRC standard mathematical tables and 
formula - p. 595 'Exponential distribution' (only a very short 
definition)]
---
---
Internet: see also:
---
Math: Probability:Distribution: Exponential: Can you derive 
Exponential distribution L . e^(-L . t)?
http://www.faqts.com/knowledge_base/view.phtml/aid/33983/fid/815
---
Exponential Distribution
http://mathworld.wolfram.com/ExponentialDistribution.html
---
Exponential Distribution
http://en.wikipedia.org/wiki/Exponential_distribution
---
Exponential Distribution: History
http://members.aol.com/jeff570/e.html
---
Exponential Distribution: History: Karl Pearson
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pearson.html
---
Math: Probability: Distribution: Link: Overview: Can you give an 
overview of links?
http://www.faqts.com/knowledge_base/view.phtml/aid/32917/fid/815
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