MTBF/MTBE: Mean time is the reciprocal of a constant event rate. Reliability: Survival probability P(X>x). Warranty threshold: A quantile chosen from an acceptable cumulative failure rate.

x=-ln(1-p)/lambda; reliability R(x)=exp(-lambda x)

1. گام 1Define p as cumulative failure probability.
2. گام 2Solve the quantile in matching units.
3. گام 3Verify pexp and discuss constant hazard.

متغیر

X is waiting time in minutes until the next support request.

انتخاب مدل

Use an exponential model only because the scenario explicitly assumes independent events at a constant rate.

پارامترها

lambda=0.3 requests per minute; p=0.95.

بیان احتمال

Find x such that P(X<=x)=0.95.

محاسبه

Solve P(X<=x)=0.95: x=-ln(0.05)/0.3=9.986 minutes. Verify with qexp(0.95, rate=0.3).

تفسیر

Solve P(X<=x)=0.95: x=-ln(0.05)/0.3=9.986 minutes. Verify with qexp(0.95, rate=0.3).

بررسی معقول‌بودن

The 95th percentile must exceed both the median ln(2)/lambda and mean 1/lambda because it lies far into the right tail.

Solve P(X<=x)=0.95: x=-ln(0.05)/0.3=9.986 minutes. Verify with qexp(0.95, rate=0.3).

متغیر

X is component life in years.

انتخاب مدل

Use an exponential model only because the scenario explicitly assumes independent events at a constant rate.

پارامترها

MTBF=5 years, so lambda=1/5=0.2 failures per year; p=0.10.

بیان احتمال

Find x such that P(X<=x)=0.10.

محاسبه

Rate lambda=1/5 per year and qexp(0.10,rate=0.2)=0.527 years, about 6.3 months. This is a policy threshold, not a lifetime guarantee.

تفسیر

Rate lambda=1/5 per year and qexp(0.10,rate=0.2)=0.527 years, about 6.3 months. This is a policy threshold, not a lifetime guarantee.

بررسی معقول‌بودن

The cutoff is shorter than the 5-year mean because it protects against only the earliest 10% of modelled failures; rising hazard would invalidate the policy calculation.

Rate lambda=1/5 per year and qexp(0.10,rate=0.2)=0.527 years, about 6.3 months. This is a policy threshold, not a lifetime guarantee.

متغیر

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

انتخاب مدل

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

پارامترها

Symbols: p is cumulative failure probability with 0<p<1; lambda>0 is failures per time unit; x, MTBE, and MTBF use the reciprocal time unit. Reliability R(x) is unit-free.

بیان احتمال

Find MTBF.

محاسبه

x=-ln(1-p)/lambda; reliability R(x)=exp(-lambda x). MTBF=1/lambda=2,000 hours; the median is lower because the exponential model is right-skewed.

تفسیر

MTBF=1/lambda=2,000 hours; the median is lower because the exponential model is right-skewed.

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.

MTBF=1/lambda=2,000 hours; the median is lower because the exponential model is right-skewed.

متغیر

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

انتخاب مدل

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

پارامترها

Symbols: p is cumulative failure probability with 0<p<1; lambda>0 is failures per time unit; x, MTBE, and MTBF use the reciprocal time unit. Reliability R(x) is unit-free.

بیان احتمال

Find P(life>500).

محاسبه

x=-ln(1-p)/lambda; reliability R(x)=exp(-lambda x). Reliability is exp(-1)=0.3679.

تفسیر

Reliability is exp(-1)=0.3679.

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.

Reliability is exp(-1)=0.3679.

متغیر

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

انتخاب مدل

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

پارامترها

Symbols: p is cumulative failure probability with 0<p<1; lambda>0 is failures per time unit; x, MTBE, and MTBF use the reciprocal time unit. Reliability R(x) is unit-free.

بیان احتمال

Assess exponential warranty analysis.

محاسبه

x=-ln(1-p)/lambda; reliability R(x)=exp(-lambda x). Constant hazard is not credible across the full horizon; compare an age-dependent reliability model.

تفسیر

Constant hazard is not credible across the full horizon; compare an age-dependent reliability model.

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.

Constant hazard is not credible across the full horizon; compare an age-dependent reliability model.

qexp() returns a time threshold from a left-tail failure probability; pexp(..., lower.tail=FALSE) returns survival. 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.

Turn target service and failure probabilities into time thresholds while challenging constant hazard. 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.