Uniform Distribution Calculator
Free uniform distribution calculator. Compute PDF, CDF, mean, variance, P(cā¤Xā¤d) for continuous and
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Uniform ā Continuous and Discrete: PDF, CDF, Mean, Variance
Equal probability for all outcomes. Fair die, random numbers, measurement error. P(cā¤Xā¤d), percentiles.
Real-World Scenarios ā Click to Load
Inputs
PDF with P(cā¤Xā¤d) Shaded
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- ā¢ Continuous Uniform(a,b): PDF f(x)=1/(b-a) for aā¤xā¤b; CDF F(x)=(x-a)/(b-a); Mean=(a+b)/2, Variance=(b-a)Ā²/12
- ā¢ Discrete Uniform {a,...,b}: P(X=k)=1/(b-a+1); Mean=(a+b)/2, Variance=((b-a+1)Ā²-1)/12
- ā¢ P(cā¤Xā¤d) = (d-c)/(b-a) for continuous; (number of integers in [c,d])/(b-a+1) for discrete
- ā¢ Percentile q: x_q = a + q(b-a) for continuous
- ā¢ All outcomes equally likely ā maximum entropy distribution on a bounded interval
Did You Know?
How It Works
1. Continuous Uniform
PDF is constant on [a,b]; area under curve = 1. CDF increases linearly from 0 to 1.
2. Discrete Uniform
Each integer k in {a,...,b} has probability 1/(b-a+1). CDF is a step function.
3. Mean and Variance
Both types share mean (a+b)/2. Variance differs: continuous (b-a)Ā²/12 vs discrete ((b-a+1)Ā²-1)/12.
4. Probability in Interval
P(cā¤Xā¤d) = length of overlap / total length. For discrete, count integers in [c,d] ā© {a,...,b}.
5. Percentiles
q-th percentile: x_q = a + q(b-a) for continuous. For discrete, use the k-th smallest value where CDF first exceeds q.
Expert Tips
Prior of Ignorance
Use uniform as uninformative prior when you only know bounds. Beta(1,1) is uniform on [0,1].
Rounding
Rounding to nearest unit creates discrete uniform on the error: e ~ U(-0.5, 0.5).
Inverse Transform
To simulate U(a,b): generate U~U(0,1), then X = a + U(b-a).
Fairness
Uniform is the "fair" distribution ā no outcome is favored. Used to model fair coins, dice, lotteries.
Why Use This Calculator vs Other Tools?
| Feature | This Calculator | Excel | R | SciPy |
|---|---|---|---|---|
| Continuous + Discrete | ā | ā ļø UNIFORM only | ā ļø runif/dunif | ā |
| PDF + CDF + P(cā¤Xā¤d) | ā | ā ļø Multiple | ā ļø Multiple | ā |
| 7 real-world presets | ā | ā | ā | ā |
| Shaded probability chart | ā | ā | ā | ā ļø Manual |
| Percentile calculator | ā | ā ļø | ā | ā |
| Educational content | ā | ā | ā | ā |
Frequently Asked Questions
What is the difference between continuous and discrete uniform?
Continuous: X can take any value in [a,b]. Discrete: X takes only integers a, a+1, ..., b. A die is discrete; bus arrival time is often modeled continuous.
When should I use the uniform distribution?
When all outcomes in a range are equally likely: fair dice, random number generation, measurement error with known bounds, prior of ignorance in Bayesian analysis.
What is the variance of Uniform(a,b)?
Continuous: (b-a)Ā²/12. Discrete on {a,...,b}: ((b-a+1)Ā²-1)/12. The continuous formula is simpler.
How do I compute P(c ā¤ X ā¤ d)?
Continuous: (d-c)/(b-a) if [c,d] ā [a,b], else use the overlap. Discrete: count integers in [c,d] ā© {a,...,b} and divide by (b-a+1).
What is the 50th percentile (median)?
For both: (a+b)/2. The median equals the mean for the uniform distribution.
Can the uniform be used as a prior?
Yes. Uniform is the maximum entropy (least informative) prior when you only know the support. Common in Bayesian inference.
How do I simulate from Uniform(a,b)?
Generate U ~ Uniform(0,1) (e.g., Math.random()), then X = a + U*(b-a).
What is the relationship to the beta distribution?
Beta(1,1) is the uniform distribution on [0,1]. Uniform is a special case of the beta.
Uniform Distribution by the Numbers
Official Data Sources
Disclaimer: This calculator uses exact closed-form formulas for the uniform distribution. Results are mathematically exact. For discrete uniform, a and b are floored to integers. This tool is for educational and professional reference purposes.
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