# 2. Uniform Distribution

One simple, basic example of a continuous random variable is one where the random variable X can take any value in a given interval with an equally likely probability. The distribution of such a random variable is the uniform distribution.

Image you show up for work one morning and are told there will be a fire alarm drill sometime during the eight-hour day. Fire drills don’t make sense if everyone knows when the drill will take place, so all you know is that sometime during the day, a drill will take place. This means that at every moment there is an equally likely chance that the fire drill will take place. Together with the information that the drill will happen, i.e., there is a 100% = 1 probability that it will occur, we get the following distribution: Why is the probability fixed at 1/8? Use the facts that (1) there are 8 hours during which the drill can take place, and (2) there is a 100% probability of the drill occurring. Since the uniform distribution is a rectangle, and the area of any rectangle is A(length× (width), we get:

1 = 8 × height

and solving for height gives us: Now you can determine the probabilities of the drill taking place during any time interval you choose. For example, the probability that the drill will occur during your lunch hour (from 12:00 p.m. to 1:00 p.m.) is simply the area of the region shown in red:  Once again, remember that AREA = PERCENTAGE OR PROBABILITY. The discussion of probability density curves always starts with the uniform distribution, because everyone knows how to calculate areas of rectangles. And it’s easy to see how the concepts of area and probability are linked.