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Feb 20th, 2005 02:35
Knud van Eeden,
  Knud van Eeden  28 December 2004  09:35 pm  Math:Probability:Distribution:Exponential: Can you describe Exponential distribution L . e^(L . t)?  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 387911065  p. 609 'Exponential distribution' (short formula)]  [book: Daintith, John / Nelson, R. David  the penguin dictionary of mathematics  ISBN 0140511199  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 0201 592789  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 047117026 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 0891165738  p. 260 'Exponential or Boltzmann distribution' (only a few short formulas)]  [book: Spiegel, Murray R.  Probability and statistics  McGrawHill (Schaum series)  p. 119 'The Exponential distribution' (only a very short formula)]  [book: Weisstein, Eric W.  CRC concise encyclopedia of mathematics  ISBN 0849396409  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://wwwgroups.dcs.stand.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 