Distribution with upper and lower boundaries

Is there any continuous distribution that has an upper level of 1 and a lower level of 0? The reason I am asking this is that I want to know the distribution of my data which shows the roll over ratio of customer deposit which might only be between 0 (means all the fund is withdrawn) and 1 (means all the fund is rolled over).

• How about a uniform distribution over the interval [0,1]? Or, more generally, a beta distribution? Oct 1, 2013 at 15:45
• Some parametric continuous distribution might provide a convenient way to summarize the data if it fits them well. But, apart from that, why do you seek such a distribution in the first place?
– whuber
Oct 1, 2013 at 17:08
• I am preparing a scenario for a stress test that with a certain confidence level i will be sure that a certain percentage of the funds deposited to our bank will be withdrawn. Do you have any suggestion which distribution will be appropriate for this requirement? Oct 1, 2013 at 21:47
• In what way does a 'stress test' require you to specify a particular distributional form? What does this test consist of? Oct 2, 2013 at 0:27
• I am constructing a scenario to determine the maximum shortage of fund in my bank. Currently i have a historic data of number of amounts of deposits which is rolled over by the depositors. By using this information, i want to create a continuous distribution of the roll over ratio, so that i can be convinced with a cetain confidence level how many percentage of the deposits will stay and how many percentage will be withdrawn Oct 3, 2013 at 13:18

Some examples of continuous pdf's defined with domain of support on (0,1) are:

Arc-Sine, Beta, Bradford, Hahn, Inverse Triangular, Kumaraswamy, Power Function, semi-circle up, semi-circle down, skew-V, standard two-sided Power, Topp-Leone, Trapezoidal, Triangular, U-quadratic, Uniform, V and so on.

For your application, you might also want to also look into censoring (or doubly censoring) the underlying random variable below at 0, and/or above at 1.

There are an infinite number of continuous distributions on the range 0 to 1.

The most obvious of the widely used families would be the beta-family, but there are many others.

However, the chances that any of the well known ones will be the distribution you actually have is zero.

In fact, I bet you don't really have a continuous distribution at all, since there will likely be a nonzero proportion of the distribution exactly at both 0 and 1. That would imply you'd be better off considering a mixture of a Bernoulli and a continuous distribution on [0,1]. You might be able to get an adequate approximation to the nominally continuous portion via say mixtures of some standard distributions, such as a mixture of betas, yielding a simple finite mixture of discrete and continuous distributions for the whole thing.

[If you're more lucky, a single continuous distribution of simple form might be a reasonable description of the part strictly between 0 and 1, but I wouldn't assume this will be the case.]