Explanation 1
Estimated time: 58 minutes.
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Lesson 8 of 11
Turn target service and failure probabilities into time thresholds while challenging constant hazard.
Lesson 7.7
Turn target service and failure probabilities into time thresholds while challenging constant hazard. Business question: Find the time by which 95% of requests arrive.
Explanation 1
Estimated time: 58 minutes.
Explanation 2
Prior knowledge: complements, algebra, probability notation, and careful unit conversion.
Explanation 3
Solve inverse exponential thresholds and connect rate to MTBE or MTBF.
Explanation 4
Design a warranty or service threshold with an explicit reliability assumption.
Explanation 5
Interpret a inverse exponential, mtbe/mtbf, reliability, and warranty result in business context with consistent units.
Explanation 6
Use and explain the relevant base R distribution functions without confusing density, probability, quantiles, or simulation.
Core explanation
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.
Explanation 1
A probability is an area over an event, not an unexplained curve height.
Explanation 2
State the random variable, support, units, and assumptions before using a formula.
Explanation 3
Connect every numerical answer to a practical decision and limitation.
Terminology
Mean time is the reciprocal of a constant event rate.
1,000 hours at rate 0.001.
Survival probability P(X>x).
Probability a unit lasts beyond warranty.
A quantile chosen from an acceptable cumulative failure rate.
10% failure before warranty expiry.
Notation and formulas
x=-ln(1-p)/lambda; reliability R(x)=exp(-lambda x)
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. Use: apply this relationship only after defining the model and event. Plain-text equivalent: x=-ln(1-p)/lambda; reliability R(x)=exp(-lambda x).
Worked example
Scenario
Support requests arrive independently at a stable rate of 0.3 per minute. Let X be the wait in minutes until the next request.
R check
qexp(0.95, rate=0.3); verify with pexp(result, rate=0.3).Solve P(X<=x)=0.95: x=-ln(0.05)/0.3=9.986 minutes. Verify with qexp(0.95, rate=0.3).
Guided practice
Scenario
A synthetic component has mean life 5 years under an assumed constant failure rate.
R check
qexp(0.10, rate=1/5); verify cumulative failures with pexp().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.
Interactive Mission
Exponential left tail
1.0000
Interval probability
11.513 time units
Independent practice
Scenario
Failures occur at rate 0.0005 per hour.
R check
qexp() returns a time threshold from a left-tail failure probability; pexp(..., lower.tail=FALSE) returns survival.MTBF=1/lambda=2,000 hours; the median is lower because the exponential model is right-skewed.
Independent practice
Scenario
A component has rate 0.002 per hour.
R check
qexp() returns a time threshold from a left-tail failure probability; pexp(..., lower.tail=FALSE) returns survival.Reliability is exp(-1)=0.3679.
Independent practice
Scenario
Observed hazard rises after year three.
R check
qexp() returns a time threshold from a left-tail failure probability; pexp(..., lower.tail=FALSE) returns survival.Constant hazard is not credible across the full horizon; compare an age-dependent reliability model.
R connection
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.
Live R Lab
Find service levels and warranty thresholds from a target probability.
Ready to run
Common mistakes and model safeguard
MTBF is a model mean, not a guaranteed lifetime; constant hazard and censoring must be considered. Also check tail direction, parameter units, support, and whether software returned density, cumulative probability, a quantile, or generated data.
Originality and provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Reflection
Name the assumption, the evidence you would seek, and how the recommendation could change.
Lesson summary
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.
Explanation 1
MTBF/MTBE: Mean time is the reciprocal of a constant event rate.
Explanation 2
Reliability: Survival probability P(X>x).
Explanation 3
Warranty threshold: A quantile chosen from an acceptable cumulative failure rate.
Assistant
Coach me through Inverse Exponential, MTBE/MTBF, Reliability, and Warranty one decision at a time. Before submission, give hints only and do not reveal quiz answers or complete my recommendation.
Resource
Download an original, accessible STATLAB Academy reference and practice sheet.
DownloadExit check
Use the short exit check after instruction and practice. Reach 70% to complete this lesson; explanations appear only after submission.
Checkpoint
Question 1 of 4. Answered 0/4. Passing score: 70%.