Independent Uniform Sample Generator
Draws independent random samples of size n = 7 from the continuous standard uniform distribution U[0, 1].
Statistics & Probability / Order Statistics & Quantiles
FREE TO EXPLORELABORATORY 09 · STATISTICS & PROBABILITY
Sample Quantiles, Median, and Order Statistics Distributions
Derive and simulate distributions of sample order statistics X_{(r)} from uniform and exponential parents, verify the Beta distribution representation of uniform order statistics, and quantify mean order spacing. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EXPECTED ORDER STATISTIC E[X_(R)]—
SAMPLING TELEMETRY
Ready to sample with Order Index r = 1.0.
Adjust Order Index r, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
Order statistics are non-linear transformations of the data, making their analytical distributions intractable for complex non-standard parent distributions. The empirical regression confirms the theoretical order spacing E[X_{(r)}] = 0.1250 · r for n = 7 uniform variates with R² = 1.000000.
Statistical tools & sampling modules.
Order Statistic Sorting Pipeline
Sorts each sample array into ascending order statistics X_{(1)} ≤ X_{(2)} ≤ ... ≤ X_{(n)}.
Beta Distribution Density Mapper
Evaluates theoretical Beta density f_{(r)}(x) = [n! / ((r-1)!(n-r)!)] x^(r-1) (1-x)^(n-r) for each rank r.
Sample Quantile Inversion Unit
Estimates continuous population quantiles using linear interpolation across order statistics.
Sample Spacing Difference Meter
Measures spacing intervals D_r = X_{(r+1)} - X_{(r)} and verifies their identical distribution.
Linear Spacing Regression Compiler
Performs linear regression of E[X_{(r)}] versus r to confirm the exact theoretical gradient m = 1/(n+1) = 0.125.
FOLLOW THE METHOD
A reason behind every step.
Derive and simulate distributions of sample order statistics X_{(r)} from uniform and exponential parents, verify the Beta distribution representation of uniform order statistics, and quantify mean order spacing. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 rank.
01
Set Sample Dimension and Parent Distribution
Specify sample size n = 7 drawn from a continuous standard uniform parent distribution U[0, 1] across M = 20,000 Monte Carlo trials.
02
Sort Sample Vectors into Order Statistics
For each simulated trial, sort the 7 uniform observations in ascending order to establish the order vector (X_{(1)}, X_{(2)}, ..., X_{(7)}).
03
Evaluate First Order Statistic (Sample Minimum)
For r = 1, compute the empirical expectation E[X_{(1)}] across all 20,000 trials, confirming agreement with theoretical Beta(1, 7) mean E[X_{(1)}] = 1/(7+1) = 0.1250.
04
Step Through Order Statistics r = 1 to 6
Evaluate the empirical mean for each order rank r ∈ {1, 2, 3, 4, 5, 6}, recording the exact step-wise increment of 1/(n+1) = 0.125 per rank.
05
Verify Beta Distribution Representation
Confirm that the r-th order statistic follows a Beta distribution Beta(r, n - r + 1), with theoretical expectation E[X_{(r)}] = r / (n + 1).
06
Execute Linear Spacing Regression
Fit a linear regression line y = mx + b to the coordinates (x = r, y = E[X_{(r)}]). Confirm that the empirical gradient m = 0.1250 exactly verifies the theoretical order spacing.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Order Statistics & Quantiles mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.1250 · x
| # | Order Statistic Index r (rank) | Expected Order Statistic E[X_(r)] (expectation) | Order Statistic Rank r | Expected Value E[X_(r)] | Remove |
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