A real-valued continuous random variable is uniformly distributed if the probability that lands in an interval is proportional to the length of that interval.
But if is infinite, say, a subinterval of , then , so defining by giving the probabilities that equals a certain element of will not work. Instead, one defines by assigning probabilities to subsets of . These probabilities are assigned by weighting subsets based on their measure. For example, when , for any , one has the probability .
The uniform distribution on is denoted , and has PDF Integrating this function, one observes the cumulative density function for is
Consider the following example computation, using this information:
Let be independent, and identically distributed . Compute the probability distribution function for the random variable .
Each has PDF for and elsewhere. Let denote the CDF for . It follows Differentiating this gives the PDF