Exponential Distribution Calculator

Exponential Distribution is evaluated from Rate Parameter - events per unit time and Time Value. The calculation reports P - Probability within time x, P - Probability beyond time x and Mean.

Results

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About the Exponential Distribution Calculator

Exponential Distribution is treated here as a quantitative relation between Rate Parameter - events per unit time and Time Value and P - Probability within time x, P - Probability beyond time x, Mean and Std Dev.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
CDF: P(X<=x) = 1 - e^( - λx)
Survival: P(X>x) = e^( - λx)
Mean = standard deviation = 1/λ

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

CDF: P(X<=x) = 1 - e^( - λx)
Survival: P(X>x) = e^( - λx)
Mean = standard deviation = 1/λ

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Customer service: Average 10 calls/hour (λ=10/60 per minute). P(next call within 4 minutes)

Inputs

lambda: 0.1667 x: 4

P - Probability within time x: 0.486651. P - Probability beyond time x: 0.513349. Mean: 5.9988. Std Dev: 5.9988. Median: 4.1581

With Rate Parameter - events per unit time = 0.1667 and Time Value = 4 as the stated inputs, the result is P - Probability within time x = 0.486651, P - Probability beyond time x = 0.513349 and Mean = 5.9988. Each value corresponds to the declared output fields.

Example 2: Equipment failure: MTBF = 2000 hours (λ = 1/2000). P(failure within 500 hours)

Inputs

lambda: 0.0005 x: 500

P - Probability within time x: 0.221199. P - Probability beyond time x: 0.778801. Mean: 2,000. Std Dev: 2,000. Median: 1,386.2944

With Rate Parameter - events per unit time = 0.0005 and Time Value = 500 as the stated inputs, the result is P - Probability within time x = 0.221199, P - Probability beyond time x = 0.778801 and Mean = 2,000. Each value corresponds to the declared output fields.

Example 3: Website server: Average 0.3 requests/second (λ=0.3). P(no request for more than 5 seconds)

Inputs

lambda: 0.3 x: 5

P - Probability within time x: 0.77687. P - Probability beyond time x: 0.22313. Mean: 3.3333. Std Dev: 3.3333. Median: 2.3105

With Rate Parameter - events per unit time = 0.3 and Time Value = 5 as the stated inputs, the result is P - Probability within time x = 0.77687, P - Probability beyond time x = 0.22313 and Mean = 3.3333. Each value corresponds to the declared output fields.

Example 4: Sales: average 1 deal per 8 days (λ = 0.125/day). P(next deal within 5 days)

Inputs

lambda: 0.125 x: 5

P - Probability within time x: 0.464739. P - Probability beyond time x: 0.535261. Mean: 8. Std Dev: 8. Median: 5.5452

With Rate Parameter - events per unit time = 0.125 and Time Value = 5 as the stated inputs, the result is P - Probability within time x = 0.464739, P - Probability beyond time x = 0.535261 and Mean = 8. Each value corresponds to the declared output fields.

Common Use Cases

Model time between events in a Poisson process
Calculate reliability and failure probability
Customer waiting time and service time analysis