Multi-Distribution Variate Generator
Generates 100,000 independent random samples from Normal, Exponential, Student's t, and Bimodal mixture distributions.
Statistics & Probability / Chebyshev & Markov Bounds
FREE TO EXPLORELABORATORY 05 · STATISTICS & PROBABILITY
Empirical Verification of Markov's and Chebyshev's Inequalities
Evaluate empirical tail probabilities across diverse probability distributions, compare empirical tail frequencies against Markov's and Chebyshev's upper bounds, and verify the inverse-square power law. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
CHEBYSHEV UPPER BOUND 1/K²—
SAMPLING TELEMETRY
Ready to sample with Multiple k = 1.5.
Adjust Multiple k, 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
Chebyshev's bound is conservative for normal data: at 3σ, normal tail is 0.27% while Chebyshev gives 11.11%. The empirical regression of ln(P_bound) versus ln(k) confirms the theoretical inverse-square concentration scaling α = -2.000 with R² = 1.000000.
Statistical tools & sampling modules.
Population Parameter Standardizer
Computes exact theoretical mean μ and standard deviation σ to construct normalized deviation scores Z = |X - μ| / σ.
Empirical Tail Event Counter
Accumulates the empirical indicator function ∑ I(|X_i - μ| ≥ kσ) to measure empirical two-sided tail probabilities.
Markov Inequality Evaluator
Evaluates Markov's bound P(X ≥ a) ≤ E[X] / a for non-negative random variables.
Chebyshev Inequality Evaluator
Calculates Chebyshev's non-parametric concentration bound P(|X - μ| ≥ kσ) ≤ 1 / k².
Log-Log Exponent Regression Engine
Performs linear regression on ln(Bound) versus ln(k) to confirm the exact quadratic exponent α = -2.000.
FOLLOW THE METHOD
A reason behind every step.
Evaluate empirical tail probabilities across diverse probability distributions, compare empirical tail frequencies against Markov's and Chebyshev's upper bounds, and verify the inverse-square power law. Standardized parameter settings: 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 k.
01
Configure Target Distribution and Sample Buffer
Select a continuous distribution with verified finite variance (e.g., standard normal N(0, 1) or standardized Exponential) and draw N = 100,000 observations.
02
Standardize Sample Deviations
For each sample observation, compute the standardized absolute deviation Z_i = |X_i - μ| / σ and sort the array to facilitate rapid tail searching.
03
Evaluate Tail Frequency for Deviation Multiple k = 1.5
Set k = 1.5. Calculate the theoretical Chebyshev upper bound P_bound = 1 / k² = 1 / (1.5)^2 = 0.4444. Count the empirical fraction of observations exceeding 1.5σ.
04
Verify Markov Bound on Squared Deviations
Confirm that applying Markov's inequality to the non-negative variable Y = (X - μ)² yields P(Y ≥ k²σ²) ≤ E[Y] / (k²σ²) = σ² / (k²σ²) = 1/k², deriving Chebyshev's inequality.
05
Step Deviation Multiple k Across 1.5 to 4.0
Increment k across standardized values k ∈ {1.5, 2.0, 2.5, 3.0, 3.5, 4.0}, recording the steep decay of both empirical tail frequencies and the theoretical 1/k² bound.
06
Execute Log-Log Regression on the Upper Bound
Fit a linear regression model to the coordinates (x = ln(k), y = ln(1/k²)). Confirm that the empirical gradient m = -2.0000 exactly verifies the inverse-square concentration scaling.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Chebyshev & Markov Bounds mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
ln(P_bound) = -2.0000 · ln(k)
| # | Standard Deviation Multiple k (k) | Chebyshev Upper Bound 1/k² (probability) | Natural Log of Deviation Multiple ln(k) | Natural Log of Upper Bound ln(1/k²) | Remove |
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