Uniform Distribution: Equal Density, Interval Probability, and Simulation

A discrete variable has countable outcomes and uses sums. A continuous variable can take any real value in an interval and assigns probability through interval areas.

A probability mass function assigns mass to discrete points. A probability density function supplies height, while integration converts that density into probability over an interval.

No. For a continuous variable, a single point has probability zero. The probability of a short interval is approximately density times width, and exact probability is area under the density curve.

No. Continuous describes the possible values; uniform describes a flat density on one interval. Normal, exponential, gamma, and beta distributions are continuous but have different shapes.

Height can vary continuously, but equal-width height ranges are not equally likely. A uniform model gives equal probability to equal-length subintervals; height data concentrate near a typical range.

The density is constant at 1/(b-a) between a and b and zero outside. Every equal-length subinterval inside the support has the same probability—not every individual real number.

For X uniform on [a,b], P(c≤X≤d)=(d-c)/(b-a) whenever a≤c≤d≤b. Endpoints do not affect continuous probabilities.

Ordinary addition handles finitely many pieces. An integral is the limit of adding increasingly thin contributions across a complete interval.

Partition [a,b] into widths Δx, add rectangle areas f(x_i)Δx, and let the maximum width approach zero. The limiting sum is the integral.

More, thinner rectangles give better area approximations. In the limit, the sum of tiny areas becomes an exact integral over the continuum.

The Fundamental Theorem of Calculus links a density-like rate to its accumulated area: differentiating an integral recovers the integrand, subject to standard continuity conditions.

At x, the squared distance is (x-μ)^2. Multiplying by f(x)dx weights that distance by the tiny interval's probability; integrating adds every contribution.

Symmetry places the mean at μ=(a+b)/2. The probability to the left and right of the midpoint is equal.

Discrete variance sums squared distances times point probabilities. Continuous variance integrates squared distances times density. The concept stays fixed while the addition rule changes.

After centering at the midpoint and averaging squared distances across the flat interval, the integral evaluates to L^2/12 where L=b-a. Shifting the interval changes the mean but not its variance.

Taking the square root of uniform variance gives σ=(b-a)/√12. Variance uses squared units; standard deviation uses the same units as X.

A seeded pseudorandom number generator repeatedly transforms an internal state into new bit patterns. The same seed reproduces the same sequence; the sequence can still have strong statistical properties.

Using s_next=(7s+5) mod 100 produces a deterministic cycle. It is useful for seeing state updates, but its short cycle and uneven behavior make it unsuitable as a real random generator.

If K is uniform on {0,…,2^53-1}, then U=K/2^53 is uniform on a grid of 2^53 representable values in [0,1). It approximates, but does not sample, every real number.

The mathematical model has a flat continuous density; software returns one of finitely many floating-point values designed to behave uniformly. Theory explains interval probabilities, while simulation lets us test them empirically.