When we throw a fair dice the scores 1, 2, 3, 4, 5, 6 are all equally likely. No other scores are possible. In a situation where each outcome is equally likely we are said to have a uniform distribution- all the possible outcomes are equally likely. It is important to know that the outcome need not be a number, though it is very useful to have one such: when you drop the toast it may land butter side up or butter side down – these two outcomes may be assumed to be equally likely (though in fact they may not be).The example just given is of a discrete uniform distribution, but continuous uniform distributions are also possible – for example, the time of delivery of a parcel which could be delivered anywhere between 9am and 5pm.
As with all probability distributions, all the probabilities add up to one, so if there are n possible outcomes, each with probability
For a continuous distribution there are an infinite number of outcomes. For the example given above of a parcel being delivered the probability of the parcel being delivered at any particular time is zero, since we can divide time itself so that the length of a time interval is zero. Instead, we take the probability of the event occurring in the interval
to
The length of this time interval is
so, there being 8 hours between 9am and 5pm, the probability of the parcel being delivered in the intervalto
is
In general if for a continuous distribution, all outcomes between a and b are equally likely and none other are possible, then probability of an occurrence in any interval of length
is
The mean of a uniform distribution is, as you might expect, in the middle of the interval, every outcome being equally likely