# Standard deviation in exponential distribution

1. May 25, 2014

### oneamp

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

2. May 25, 2014

### mathman

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.

3. May 25, 2014

### oneamp

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

4. May 25, 2014

### FactChecker

Yes.
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.

5. May 26, 2014

### oneamp

Thank you. What is the point of having 'standard distributions' with different probabilities for X contained therein? What is 'standard' about that?

6. May 26, 2014

### FactChecker

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.

7. May 26, 2014

### oneamp

Thank you very much