Standard deviation in exponential distribution

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Discussion Overview

The discussion centers around the significance of the standard deviation in the exponential distribution, particularly in comparison to the standard deviation in the normal distribution. Participants explore the implications of the standard deviation being equal to the mean in the context of probability and distribution characteristics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants inquire about the significance of the standard deviation in the exponential distribution and how it compares to the normal distribution, particularly regarding the '68-95-99.7' rule.
  • One participant suggests that the standard deviation indicates the spread of probabilities for all distributions, but the exact percentages are only applicable to the normal distribution.
  • There is a question about whether the standard deviation for an exponential distribution is indeed the same as the mean, and if this leads to specific interpretations of the range around the mean.
  • Another participant confirms that the first standard deviation for an exponential distribution can be expressed as a range around the mean, but notes that the probabilities differ significantly from those in the normal distribution.
  • Concerns are raised about the concept of 'standard distributions' when different distributions yield different probabilities for values contained within one standard deviation.
  • One participant argues that despite the differences in probabilities, standard distributions still provide valuable information about the average deviation from the mean.

Areas of Agreement / Disagreement

Participants express varying views on the implications of the standard deviation in different distributions, with some agreeing on its significance while others question the concept of 'standard distributions' due to differing probabilities. The discussion remains unresolved regarding the interpretation of these differences.

Contextual Notes

Participants reference the '68-95-99.7' rule specific to the normal distribution, highlighting the lack of similar exact probabilities for the exponential distribution. There is also mention of the need for clarity on what constitutes a 'standard distribution' given the variability in probabilities across different distributions.

oneamp
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What is the significance of the standard deviation (equal to the mean) in an exponential distribution? For example, as compared to the standard deviation in the normal distribution, which conforms to the '68-95-99.7' rule?
thanks
 
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Qualitatively the standard deviation has the same role for all distributions, indicating the spread covering most of the probability. For the normal distribution, the percentages covered are exact, while for others it gives a rough idea.
 
Thank you. Additionally, is it true that the standard deviation for an exponential distribution is the same as the mean? Does this imply that the first standard deviation for an exponential distribution is, on either side of the mean 'u', 0 - u, and u - 2u?

Thank you again
 
oneamp said:
Thank you. Additionally, is it true that the standard deviation for an exponential distribution is the same as the mean?
Yes.
Does this imply that the first standard deviation for an exponential distribution is, on either side of the mean 'u', 0 - u, and u - 2u?
Yes, 0<X<2u would be the range of X that represents "within one standard deviation of the mean". The probabilities are quite different from the '68-95-99.7' rule of the normal distribution. For the exponential, the probability of 0<X<2u is 1-e^-2 = 0.865, not 0.68. The probabilities of 1, 2, and 3 standard deviations from the mean are 0.865, 0.95, and 0.98, respectively.
 
Thank you. What is the point of having 'standard distributions' with different probabilities for X contained therein? What is 'standard' about that?
 
Although the standard distributions of different probability density functions do not give the same probabilities of X being within one standard distribution, they still have a very good use. They tell you how much you can expect, on average, the value of X to differ from its mean. So it gives the best single-number indication of how much the values of X are spread out.
 
Thank you very much
 

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