Uniform Distribution PDF/PMF Value Calculator

Tip: use Add to Table after each run, then export.

Click Load to populate the form with the row values.

Continuous Uniform, $X \\sim \\mathrm{Unif}(a,b)$ with $a < b$:

• PDF: $f_X(x)=\\begin{cases}\\dfrac{1}{b-a}, & a \\le x \\le b \\\\ 0, & \\text{otherwise}\\end{cases}$
• Mean: $\\mathbb{E}[X]=\\dfrac{a+b}{2}$
• Variance: $\\mathrm{Var}(X)=\\dfrac{(b-a)^2}{12}$

Discrete Uniform on integers $m, m+1,\\dots,n$ with $m \\le n$:

• PMF: $p_X(k)=\\begin{cases}\\dfrac{1}{n-m+1}, & k \\in \\{m,\\dots,n\\} \\\\ 0, & \\text{otherwise}\\end{cases}$
• Mean: $\\mathbb{E}[X]=\\dfrac{m+n}{2}$
• Variance: $\\mathrm{Var}(X)=\\dfrac{(n-m+1)^2-1}{12}$

Endpoint convention: For the continuous case, the value on the boundary has the same constant PDF as the interior.

1. Choose Continuous or Discrete from the Model menu.
2. Enter bounds and the evaluation point. Ensure $a<b$ or $m\\le n$.
3. Click Compute Value to see PDF/PMF, mean, and variance.
4. Click Add to Table to accumulate multiple scenarios.
5. Use Download CSV or Download PDF to export results.

For discrete mode, k must be an integer in $[m,n]$ when integer enforcement is enabled.

PDF (continuous): For $X\sim \mathrm{Unif}(a,b)$, the value $f_X(x)$ is the height of the density at $x$. It equals $\tfrac{1}{b-a}$ when $a\le x\le b$ and $0$ otherwise. A PDF is not a probability at a single point; probabilities come from areas: $\mathbb{P}(x_1\le X\le x_2)=\int_{x_1}^{x_2} f_X(x)\,dx$.

PMF (discrete): For $X$ uniform on the integers $\{m,\dots,n\}$, the value $p_X(k)$ is the probability of $X=k$. It equals $\tfrac{1}{n-m+1}$ when $k$ is an integer in $[m,n]$, and $0$ otherwise.

Units: PDF has units of $1/\text{(units of }X\text{)}$; PMF is dimensionless. Endpoint handling for the continuous case is conventional; this tool uses $[a,b]$ inclusion by default.

Cumulative Distribution Function (CDF) gives the probability that a random variable is less than or equal to a value.

Continuous Uniform, $X\sim \mathrm{Unif}(a,b)$: $$ F_X(x)=\begin{cases} 0,& xb \end{cases} $$

Discrete Uniform on integers $\{m,\dots,n\}$: $$ F_X(x)=\begin{cases} 0,& xn \end{cases} $$ (At integer $k$, $F_X(k)=\dfrac{k-m+1}{n-m+1}$.)

CDFs are monotone non-decreasing and always range from 0 to 1.

These examples illustrate how CDFs accumulate probability up to a point.

Key similarity (continuous case): symmetry about the mean. Both the continuous uniform $\\mathrm{Unif}(a,b)$ and the normal $\\mathcal{N}(\\mu,\\sigma^2)$ are symmetric around their center: the midpoint $\\tfrac{a+b}{2}$ for uniform and the mean $\\mu$ for normal. Formally, $$f\\big(\\text{center}+d\\big)=f\\big(\\text{center}-d\\big).$$ For uniform, $\\text{center}=\\tfrac{a+b}{2}$; for normal, $\\text{center}=\\mu$.

• Two-parameter families: Uniform uses $(a,b)$; normal uses $(\\mu,\\sigma)$. Each pair fully determines the distribution.
• Area under the curve gives probability: For both, probabilities are obtained by integrating the density over intervals.
• Finite mean and variance: $\\mathbb{E}[X]=\\tfrac{a+b}{2}$, $\\mathrm{Var}(X)=\\tfrac{(b-a)^2}{12}$ for uniform; $\\mathbb{E}[X]=\\mu$, $\\mathrm{Var}(X)=\\sigma^2$ for normal.
• CLT connection: The sample mean of many i.i.d. draws from either distribution is approximately normal (Central Limit Theorem).

Note: The discrete uniform shares the two-parameter and finite-moment properties, but its PMF is not continuous.

For any x outside [a,b], the PDF value is 0.

When the support has one integer, the PMF at that integer equals 1.

Formulas: mean $(a+b)/2$ or $(m+n)/2$; variance $(b-a)^2/12$ or $\big((n-m+1)^2-1\big)/12$.