Geometric Distribution Calculator
Free geometric distribution calculator. P(X=k), P(Xโคk), P(X>k), P(aโคXโคb). PMF bar chart, CDF step ch
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Geometric Distribution โ Trials Until First Success
PMF, CDF, mean 1/p, memoryless property. Dice until 6, coin until heads, sales calls until close.
Real-World Scenarios โ Click to Load
Calculation Mode
Inputs
PMF Bar Chart โ P(X=k)
CDF Step Chart โ P(Xโคk)
Survival Function โ P(X>k)
Memoryless Property Demo
P(X > s+t | X > s) = P(X > t). Given s=2 failures, additional trials until success = starting fresh.
โ Values match!
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- โข Geometric models trials until the first success in repeated Bernoulli trials
- โข PMF: P(X=k) = (1-p)^(k-1) ร p โ k failures then one success
- โข CDF: P(Xโคk) = 1 - (1-p)^k; Survival: P(X>k) = (1-p)^k
- โข Mean = 1/p, Variance = (1-p)/pยฒ, Mode = 1 (always)
- โข Memoryless: P(X > s+t | X > s) = P(X > t) โ past failures don't affect future
Did You Know?
How It Works
Bernoulli Trials
Each trial is independent with success probability p. We count trials until the first success.
The PMF
P(X=k) = (1-p)^(k-1) ร p โ (k-1) failures, then one success. k โฅ 1.
The CDF
P(Xโคk) = 1 - (1-p)^k. Complement of "all k trials fail."
Memoryless Property
Given you've failed s times, distribution of additional trials until success = starting fresh. P(X > s+t | X > s) = P(X > t).
Mean and Variance
E(X) = 1/p (higher p โ fewer trials). Var(X) = (1-p)/pยฒ. Small p โ large variance.
Expert Tips
Geometric vs Binomial
Geometric: trials until first success. Binomial: number of successes in n fixed trials.
Exponential Connection
Geometric is the discrete analog of exponential โ both are memoryless.
Median
Median = โ-1/logโ(1-p)โ. For p=0.5, median=1; for p=0.1, medianโ7.
Range Probability
P(aโคXโคb) = CDF(b) - CDF(a-1) or sum P(X=k) for k=a..b.
Why Use This Calculator vs Other Tools?
| Feature | This Calculator | Excel | R/TI-84 |
|---|---|---|---|
| PMF + CDF + Survival charts | โ | โ ๏ธ Manual | โ ๏ธ Limited |
| 7 real-world presets | โ | โ | โ |
| Range P(aโคXโคb) | โ | โ ๏ธ Manual | โ ๏ธ Limited |
| Memoryless property demo | โ | โ | โ |
| Copy & share & AI | โ | โ | โ |
Frequently Asked Questions
What is the geometric distribution?
It models the number of trials until the first success in repeated independent Bernoulli trials with constant success probability p.
What is the memoryless property?
P(X > s+t | X > s) = P(X > t). Given you've failed s times, the distribution of additional trials until success is the same as starting from scratch.
Why is the mode always 1?
P(X=1) = p is the highest for any single value when p > 0. Each subsequent trial has lower probability (1-p)^(k-1) ร p.
When geometric vs binomial?
Geometric: "How many trials until first success?" Binomial: "How many successes in n fixed trials?"
What is the expected value?
E(X) = 1/p. For p=0.2, expect 5 trials on average until first success.
How to compute P(a โค X โค b)?
Sum P(X=k) for k=a to b, or use CDF(b) - CDF(a-1) where CDF(0)=0.
Relationship to negative binomial?
Geometric is negative binomial with r=1 (trials until 1st success).
Relationship to exponential?
Geometric is the discrete analog of the exponential distribution โ both are memoryless.
Geometric by the Numbers
Official Data Sources
Disclaimer: This calculator provides geometric distribution probabilities for educational and professional reference. Assumes independent Bernoulli trials with constant p. Verify against established statistical software for critical applications.
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