Standard deviation in exponential distribution

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The standard deviation in an exponential distribution is equal to the mean, indicating the spread of values around the mean. Unlike the normal distribution, which follows the '68-95-99.7' rule, the exponential distribution has different probabilities for values within one standard deviation, with approximately 86.5% of values falling between 0 and 2u. This highlights that while standard distributions differ in their probability densities, they still effectively convey how much values can vary from the mean. The concept of standard deviation remains significant across distributions, providing a useful measure of variability. Understanding these differences is crucial for applying statistical concepts accurately.
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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