The Uniform Distribution and Geometric Probability
9 min read
The uniform distribution is the continuous world's fair die: no value in its range is preferred over any other. It is the simplest continuous distribution, the natural model for "a random point on an interval", and the doorway to geometric probability, where probabilities become lengths, areas, and volumes.
A variable has constant density on and zero density elsewhere. Every subinterval of equal length carries equal probability.
Uniform moments
The mean sits at the midpoint by symmetry. The 12 in the variance is a standard integration result worth memorising: the variance of $\mathrm{U}(0,1)$ is $1/12$.
The CDF is a straight ramp, on the interval, and interval probabilities are just relative lengths: for , the probability of landing in is . Percentiles read off immediately, the median of is the midpoint, and the 90th percentile sits nine tenths of the way along.
Geometric probability
When outcomes are uniform over a region, probability questions become geometry questions: the probability of an event is the measure, length, area, or volume, of its favourable region divided by the measure of the whole. This is the classical definition of probability with counting replaced by measuring, and it turns a family of hard-looking problems into pictures.
Geometric probability
Valid exactly when the outcome is uniform over the region, the continuous analogue of equally likely outcomes.
Two people each arrive at a uniformly random time in a one-hour window, independently, and each waits 15 minutes. Do they meet? The outcome is a point uniform on a square, and they meet when .
The miss region is two right triangles of leg 45, together a square's worth of area. Drawing the square is the entire solution; the algebra just reads the picture.
Pick uniformly in the unit square. The probability that is the area below the diagonal:
And is the area under a parabola, . Any event about two independent uniforms is a region in the square, and its probability is that region's area.
Common trap
Geometric probability requires genuine uniformity, and "random" without qualification can hide several different uniform choices. A famous paradox asks for the probability that a random chord of a circle beats the side of the inscribed triangle, and gets three different answers from three reasonable definitions of "random chord". Always pin down WHAT is uniform before measuring anything.
One more workhorse fact: for independent uniforms, questions about relative order are pure symmetry. Two independent variables satisfy , and of them fall in any particular order with probability . These order questions grow into the order statistics module later in this course.
Quick check
X is uniform on [0, 10]. What is ? Answer as a decimal.
Quick check
X and Y are independent uniforms on [0, 1]. What is the probability both exceed 0.5? Answer as a decimal.