Rate lambda: Expected events per unit of exposure. Mean wait: The reciprocal 1/lambda. Memorylessness: Remaining wait does not depend on elapsed wait under the model.

P(X<=x)=1-exp(-lambda x); P(X>x)=exp(-lambda x)

1. گام 1Align the rate and time units.
2. گام 2Choose left, right, or interval form.
3. گام 3Interpret under a constant-rate assumption.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use exponential tails only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: X>=0 is waiting time; lambda>0 is events per time unit; x uses the matching time unit. Density has reciprocal-time units; probabilities have no units.

بیان احتمال

Find P(wait>6).

محاسبه

P(X<=x)=1-exp(-lambda x); P(X>x)=exp(-lambda x). Use exp(-0.25*6).

تفسیر

Use exp(-0.25*6).

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Use exp(-0.25*6).

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use exponential tails only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: X>=0 is waiting time; lambda>0 is events per time unit; x uses the matching time unit. Density has reciprocal-time units; probabilities have no units.

بیان احتمال

Find P(wait<=10 minutes).

محاسبه

P(X<=x)=1-exp(-lambda x); P(X>x)=exp(-lambda x). Convert 10 minutes to 1/6 hour before calculating.

تفسیر

Convert 10 minutes to 1/6 hour before calculating.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Convert 10 minutes to 1/6 hour before calculating.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use exponential tails only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: X>=0 is waiting time; lambda>0 is events per time unit; x uses the matching time unit. Density has reciprocal-time units; probabilities have no units.

بیان احتمال

Find P(2<wait<5).

محاسبه

P(X<=x)=1-exp(-lambda x); P(X>x)=exp(-lambda x). Subtract exponential CDF values.

تفسیر

Subtract exponential CDF values.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Subtract exponential CDF values.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use exponential tails only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: X>=0 is waiting time; lambda>0 is events per time unit; x uses the matching time unit. Density has reciprocal-time units; probabilities have no units.

بیان احتمال

Assess one exponential model for the whole day.

محاسبه

P(X<=x)=1-exp(-lambda x); P(X>x)=exp(-lambda x). A changing rate violates the constant-rate assumption.

تفسیر

A changing rate violates the constant-rate assumption.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

A changing rate violates the constant-rate assumption.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use exponential tails only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: X>=0 is waiting time; lambda>0 is events per time unit; x uses the matching time unit. Density has reciprocal-time units; probabilities have no units.

بیان احتمال

Interpret the model's remaining wait.

محاسبه

P(X<=x)=1-exp(-lambda x); P(X>x)=exp(-lambda x). Under the model it has the same distribution as a fresh wait.

تفسیر

Under the model it has the same distribution as a fresh wait.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Under the model it has the same distribution as a fresh wait.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use exponential tails only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: X>=0 is waiting time; lambda>0 is events per time unit; x uses the matching time unit. Density has reciprocal-time units; probabilities have no units.

بیان احتمال

State the exponential mean wait.

محاسبه

P(X<=x)=1-exp(-lambda x); P(X>x)=exp(-lambda x). The rate is 0.2 per minute, so mean wait is 5 minutes.

تفسیر

The rate is 0.2 per minute, so mean wait is 5 minutes.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

The rate is 0.2 per minute, so mean wait is 5 minutes.

dexp() returns density, pexp() cumulative probability, qexp() a time quantile, and rexp() random waiting times. Run the original starter code, inspect every printed intermediate value, then modify one assumption and explain the decision impact. Keep code, formulas, function names, column names, and numerical output left-to-right.

Name the assumption, the evidence you would seek, and how the recommendation could change.

Model waiting time between stable independent events and calculate operational tail risk. The reliable workflow is: define X and units, justify the model, state the event, calculate, verify, interpret, and disclose limitations.

Use the short exit check after instruction and practice. Reach 70% to complete this lesson; explanations appear only after submission.