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Exponential Distribution Calculator

Free exponential distribution calculator. PDF, CDF, P(X≤x), P(a≤X≤b), mean, variance, percentiles. Memoryless property. Time between events in a Poisson process.

Concept Fundamentals
1.0000
Mean
63.21%
P(X≤x)
86.47%
P(a≤X≤b)
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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

⏱️
STATISTICSDistributions

Exponential Distribution — PDF, CDF, Mean, Memoryless Property

Time between events in a Poisson process. Call center waits, light bulb life, machine failures. The only continuous distribution with the memoryless property.

Geometric (discrete) →Poisson →

Real-World Scenarios — Click to Load

Inputs

exp_results.sh
CALCULATED
$ exp_dist --lambda=1 --x=1 --a=0 --b=2
Mean (1/λ)
1.0000
Variance
1.0000
P(X≤1)
63.2121%
P(0≤X≤2)
86.4665%
Median
0.6931
PDF at x
0.367879
CDF at x
0.632121
95th %ile
2.9957
Share:
Exponential Distribution Exp(λ=1)
P(X≤1) = 63.21%
P(0≤X≤2) = 86.47%
Mean = 1.000Variance = 1.000Memoryless ✓
numbervibe.com/calculators/statistics/exponential-distribution-calculator

PDF with P(a≤X≤b) Shaded

Summary Metrics

Memoryless Property Demo

P(X > s+t | X > s) = P(X > t). Given you've waited s=0.50 units, remaining wait is still Exp(λ).

P(X > 0.5+1 | X > 0.5)
36.7879%
P(X > 1)
36.7879%

✅ Values match — memoryless property verified!

Calculation Breakdown

SUMMARY
Mean
1.0000
1/λ = 1/1
Variance
1.0000
1/λ² = 1/1²
Median
0.6931
ln(2)/λ ≈ 0.693/λ
P(X≤x)
63.2121%
1 - e^(-λx) = 1 - e^(-1×1)
PROBABILITY
P(a≤X≤b)
86.4665%
e^(-λa) - e^(-λb)

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • Exp(λ) models time between events in a Poisson process — the only continuous distribution with the memoryless property
  • • PDF: f(x) = λe^(-λx) for x ≥ 0; CDF: F(x) = 1 - e^(-λx)
  • • Mean = 1/λ, Variance = 1/λ², Median = ln(2)/λ ≈ 0.693/λ
  • • P(a ≤ X ≤ b) = e^(-λa) - e^(-λb) — no integration needed
  • • Memoryless: P(X > s+t | X > s) = P(X > t) — remaining lifetime has the same distribution

Did You Know?

⏱️The exponential is the continuous analog of the geometric — both model "waiting for the first success."Source: NIST
🚌Bus arrival times are often modeled as exponential when arrivals are random (Poisson process).Source: Queueing theory
💡Light bulb lifetimes are commonly modeled with exponential (constant failure rate).Source: Reliability engineering
🧬Radioactive decay follows exponential — half-life is ln(2)/λ.Source: Physics
📞Call center hold times and inter-arrival times are classic exponential applications.Source: Operations research
🔧MTBF (Mean Time Between Failures) = 1/λ for exponential failure times.Source: IEEE

How It Works

Poisson Connection

If events occur at rate λ per unit time, the time between consecutive events is Exp(λ).

PDF and CDF

PDF f(x)=λe^(-λx) gives density; CDF F(x)=1-e^(-λx) gives P(X≤x). Closed form — no numerical integration.

Mean and Variance

Mean = 1/λ (average wait), Variance = 1/λ². Higher λ → shorter waits.

Memoryless Property

Given you've waited s units, remaining wait is still Exp(λ). Batteries and bulbs don't "remember" usage.

Percentiles

p-th percentile: x = -ln(1-p)/λ. Median = ln(2)/λ ≈ 0.693/λ.

Expert Tips

Check Constant Rate

Exponential assumes constant hazard. If failure rate increases with age, use Weibull.

λ Units Matter

λ must match time unit. λ=2/hr means mean wait = 0.5 hr = 30 min.

Exponential vs Gamma

Exponential is Gamma(1, λ). Sum of k exponential waits → Gamma(k, λ).

Queueing Formulas

M/M/1 queue assumes exponential inter-arrival and service times.

Why Use This Calculator vs Other Tools?

FeatureThis CalculatorExcelR/SciPy
PDF + CDF + Percentiles✅⚠️ Multiple⚠️ Multiple
P(a≤X≤b) shaded region✅❌⚠️ Manual
7 real-world presets✅❌❌
Memoryless property demo✅❌❌
Interactive charts✅❌⚠️ Code
Copy & share & AI✅❌❌

Frequently Asked Questions

What is the memoryless property?

P(X > s+t | X > s) = P(X > t). If you've waited s units, remaining wait has the same Exp(λ) distribution. The past doesn't affect the future.

When should I use exponential?

When modeling time between events in a Poisson process: call arrivals, bus arrivals, machine failures (constant rate), radioactive decay.

Relationship to Poisson?

Poisson(λt) counts events in time t; Exp(λ) is time until first event. If N~Poisson(λt), time until first event is Exp(λ).

How do I interpret λ?

λ is the rate (events per unit time). Mean wait = 1/λ. λ=2/hr means average 30 min between events.

Why for reliability?

When failure rate is constant (no wear-out), time to failure is exponential. MTBF = 1/λ.

Can exponential have negative values?

No. Support is x ≥ 0. Zero for x < 0.

What is the median?

Median = ln(2)/λ ≈ 0.693/λ. Always less than mean (1/λ) because right-skewed.

How to fit to data?

Estimate λ = 1/sample_mean. MLE: λ̂ = n/Σx_i. Check constant rate with hazard plot.

Exponential by the Numbers

1/λ
Mean
1/λ²
Variance
ln(2)/λ
Median
Memoryless
Unique Property

Disclaimer: This calculator uses exact closed-form formulas. For critical applications (reliability, queueing), verify assumptions (constant rate, Poisson process). Educational and professional reference only.

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