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Geometric Distribution

Geometric distribution calculator. Enter p and k to compute first-success probability, mean and variance, with PMF notes—for discrete probability.

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P(X=k)=(1−p)^(k−1)·p
0
compute P(X = k)

📖 Tutorial | Geometric Distribution

1. Definition
Geometric distribution models the number X of trials until the first success: P(X=k)=(1−p)^(k−1)·p (k−1 failures then success).
2. Symbols
SymbolMeaning
psuccess probability per trial
ktrials to first success
E(X)=1/pmean trials
Var(X)=(1−p)/p²variance
3. How it works
  • pmf P(X=k)=(1−p)^(k−1)·p;
  • Mean E(X)=1/p;
  • Variance Var(X)=(1−p)/p²;
  • Requires 0
4. Steps
  1. Enter p, e.g. 0.2;
  2. Enter k, e.g. 3;
  3. Click Calculate for P(X=3);
  4. Read the mean and variance.
5. Example
Example: p=0.2, find P(X=3). =(0.8)²×0.2=0.128; mean 1/0.2=5.
6. Pitfalls
k must be an integer ≥1;
p must be in (0,1];
This is “trials to first success”, not “k successes”.

❓ FAQ | Geometric Distribution

Why is the mean 1/p?
With success prob. p per trial, you need on average 1/p trials.
Relation to the binomial?
Geometric: trials to first success; binomial: total successes in n trials.
Can k be 0?
No; at least one trial is needed.
Is it memoryless?
Yes; remaining trials are independent of past failures.
Where is it used?
Shooting until first hit, first defective item drawn.