Geometric Distribution Moments Mean Variance Calculator

Rigorous geometric distribution moments tool for students analysts and researchers. Enter a success probability choose the trial convention and instantly view mean variance and second moment with formulas validation and tips. Built for clarity precision and reproducible study plus quick checks for teaching lab and homework. Optimized for desktop tablet mobile with accessible controls.

Calculator Inputs

Enter a value greater than 0 and at most 1

Core moment formulas

Trials until first success support 1 2 3 ...

  • Mean E[X] = 1 ⁄ p
  • Variance Var(X) = (1 - p) ⁄ p2
  • Second moment E[X2] = (2 - p) ⁄ p2

Failures before first success support 0 1 2 ...

  • Mean E[X] = (1 - p) ⁄ p
  • Variance Var(X) = (1 - p) ⁄ p2
  • Second moment E[X2] = (1 - p)(2 - p) ⁄ p2

Results

Enter p and choose a convention then press Compute to see results

Understanding the parameter p

This calculator evaluates moments for the geometric distribution focusing on mean variance and the second raw moment. The model uses one parameter p that represents the probability of success in a single independent trial. Two conventions are supported. The first sets X as the number of trials until the first success and the support begins at one. The second sets X as the number of failures before the first success and the support begins at zero.

Supported conventions trials or failures

Choose trials until first success when you count trials from one or failures before first success when you count failures from zero.

How to use this calculator

Select a convention set p and run the computation then copy or print the results for your notes.

Core moment formulas

Formulas above match the selected convention and update your interpretation of mean variance and the second moment.

Practical interpretation

Use the mean to plan typical effort and use the variance and standard deviation to gauge range and uncertainty.

Validation rounding and precision

Input checks prevent invalid p and the display uses stable rounding while internal math keeps higher precision.

Frequently asked questions

What does p mean here

It is the probability of success in one independent trial.

Which convention should I select

Select trials until first success if you count attempts and select failures before first success if you count failures.

Can p be zero or one

p must be greater than zero and can equal one in which case the mean becomes one or zero depending on convention.

Why do means differ

The two definitions shift the support so counting trials starts at one while counting failures starts at zero.

What are the units

Units are counts of trials or failures depending on your convention.

Does this model assume independence

Yes trials must be identical and independent for geometric formulas to hold.

How are values rounded

Results are shown to six decimals while internal steps keep higher precision.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.