Uniform Random Number Distribution & Probability Theory
The Random Number Generator implements a discrete uniform distribution $U[a, b]$ over the closed integer interval $[a, b]$. In a fair discrete uniform distribution, every integer $k$ within the interval possesses an identical probability of generation:
P(X = k) = \frac{1}{b - a + 1}
The theoretical expectation (mean) and variance of the distribution are formulated as:
\mathbb{E}[X] = \frac{a + b}{2}, \quad \text{Var}(X) = \frac{(b - a + 1)^2 - 1}{12}
Scientific, Engineering & Everyday Applications
- Monte Carlo Simulations: Generating stochastic inputs for financial risk modeling and scientific physics simulations.
- Lotteries & Raffles: Generating non-repeating winning ticket numbers without manual bias.
- Scientific Sampling: Selecting randomized subjects or data records for unbiased clinical and sociological trials.
- Game Development & Tabletop Gaming: Simulating dice rolls ($1–6$, $1–20$, $1–100$) and random loot drops.
Frequently Asked Questions
How many numbers can I generate at once?
You can generate up to 10,000 numbers in a single execution. Results can be copied immediately or downloaded as a standard CSV spreadsheet.
What happens if the requested quantity exceeds the range when duplicates are disabled?
When "Allow duplicate numbers" is unchecked, the quantity cannot exceed the total integers available in the range $(b - a + 1)$, ensuring no impossible configurations.