A little help understanding the exponential distribution

[Eutrophicati]

What exactly is the difference between the mean waiting time and the median waiting time for an exponential distribution? I'm looking for a slightly intuitive understanding. I know the formulae, with the mean waiting time as 1/λ and the median as ln2/λ (which I notice is also the formula for half-life in radioactive decay), but I'm still trying to grasp the essence of what these values actually mean.

Does anyone have some useful analogies? I don't like solving these problems by plugging in values without a proper understanding of the method.

 

[mathman]

For any distribution, the mean is the average, while the median is the point at which the cumulative probability is 1/2.

Specifically, let X be a random variable with F(x) as the distribution function.

mean = ∫xdF(x) (= ∫xf(x)dx, when f(x) (= probability density) exists)

median = x value where F(x) = 1/2.

 

[camillio]

I'd add to mathman's post that the median is a robust statistical measure of central tendency while the mean value is not. This means, e.g., that median is not sensitive to outliers. A popular example considers of salaries of even number of people, e.g.: {2, 2, 3, 4, 4, 5, 22}. This set has mean value 6, although with only one exception there are values lower than 6. On the other hand, the median is 4 and it quite nicely expresses something intuitively more "central" than the mean. Try to think about some examples of variables obeying the exponential distribution.

 

[chiro]

To add to camillio's point about the mean being a misleading characteristic of a population as opposed to the median, think about things like wealth statistics where you have some very high figures on the very top end that end up skewing the mean in a drastic way.

The median however would represent the figure a lot more accurately in a way that the mean would not.

For this kind of reason, median values are good especially when you have dramatically skewed distributions.

 

[camillio]

Yet to add a little more information to Eutrophicati... :-)) Although the median has quite appealing properties, its main difficulty consists in nontrivial computations compared to the mean value, especially in Bayesian statistics. Evaluation of the mean is usually trivial, while the median is often more tricky, e.g. it needs data sorting and slicing etc. There is a field of statistics, called robust statistics, extensively using the robust measures.

 

[viraltux]

Hi,

OK, for an intuitive explanation of what is the exponential distribution is I think it is useful to think about the Poisson distribution first.

The Poisson distributions counts equally likely random events in a given framework (period of time, surface, etc...) Since you're talking about time, just imaging the number of phone calls received in a call center within an hour.

The Exponential distribution would measure the time between calls, therefore in this example $1/\lambda$ is the average time between calls which is a useful information if you want to estimate $\lambda$ (better than the median since the mean has a lower variance)

The median gives you a better picture of what is the time that eventually is going to take for the next phone call to come in since it pretty much ignores the big values given by the Exponential distribution.

For most practical purposes medians are the way to go... means are evil... and mean.