Inputs
Formulas
For a discrete uniform distribution on the integers \( \{a, a+1, \dots, b\} \) with support size \( n = b-a+1 \):
- Mean: \( \displaystyle \mu = \frac{a+b}{2} \)
- Variance: \( \displaystyle \sigma^2 = \frac{n^2 - 1}{12} \)
- Standard deviation: \( \displaystyle \sigma = \sqrt{\frac{n^2 - 1}{12}} \)
- Raw moment of order \(k\): \( \displaystyle \mathbb{E}[X^k] = \frac{1}{n}\sum_{x=a}^{b} x^k \)
- MGF: \( \displaystyle M_X(t) = \begin{cases} \dfrac{e^{ta}\big(1 - e^{tn}\big)}{n \big(1 - e^{t}\big)}, & t \neq 0 \\[6pt] 1, & t = 0 \end{cases} \)
Results
Enter valid inputs and click Calculate to see results here.
Quality checks
- When \( a=b \), the variance should be 0 and the mean equals \( a \).
- For symmetric intervals (e.g., \(-m\) to \(m\)), the mean should be 0.
- The MGF at \( t=0 \) equals 1.
FAQs
It is the set of equally likely integers from \(a\) to \(b\), inclusive, so there are \(n=b-a+1\) points. Each point has probability \(1/n\).
The points are symmetric around the midpoint of the interval. Pairing the smallest with the largest, each pair averages to \((a+b)/2\), so the overall average equals that midpoint.
Using \( \mathrm{Var}(X)=\mathbb{E}[X^2]-\mu^2 \) and Faulhaber sums for \( \sum x \) and \( \sum x^2 \), one obtains \( \sigma^2=(n^2-1)/12 \) where \( n=b-a+1 \).
The distribution is degenerate at \(a\). The mean is \(a\), the variance is 0, and all raw moments equal \(a^k\).
For small \(t\), both numerator and denominator involve \(1-e^{t}\). Using
expm1 avoids catastrophic cancellation and yields stable results close to \(t=0\).
No. This calculator assumes a finite set of integer outcomes with equal probability. The continuous uniform distribution has a continuum of outcomes over an interval and different moment formulas.
Yes. Use the raw moment field for any integer \(k\ge 1\). Central moments follow from raw moments using binomial expansions, e.g., \( \mu_2=\mathrm{Var}(X) \), \( \mu_3=\mathbb{E}[(X-\mu)^3] \), etc.