Exam P Exponential Distribution Practice Problems

Try this original problem before opening the solution. Then compare the method with SOA Exam P sample question 28.

Original practice · fully worked

Original variant: conditional survival after an elapsed operating time

The operating time T of a sensor follows an exponential distribution. Calibration shows that P(T exceeds 4 hours)=0.80. Given that the sensor is still operating at hour 4, determine the probability that it remains operating beyond hour 10.

A
0.64000
B
0.70000
C
0.80000
D
0.84000
E
0.71554

Show the answer and worked solution

Answer and calculation

Memorylessness turns the condition into six additional hours of survival. Scaling the four-hour survival gives 0.80⁽⁶⁄⁴⁾=0.71554, choice E.

Setup

Record that the exponential survival probability through four hours is 0.80.

e^{-4\lambda}=0.80

Model

By memorylessness, conditional survival from hour 4 to hour 10 is the unconditional probability of surviving six additional hours.

\Pr(T>10\mid T>4)=\Pr(T>6)=e^{-6\lambda}

Compute

Scale the four-hour survival to six hours by raising 0.80 to the 6/4 power, giving 0.715542.

e^{-6\lambda}=(e^{-4\lambda})^{6/4}=0.80^{1.5}=0.715542

Answer

The conditional probability of operating beyond hour 10 is 0.71554, which is choice E.

\boxed{0.71554\quad\text{(E)}}

Technique orientation

What exponential distribution questions are testing

The exponential model is a one-parameter waiting-time distribution with a constant hazard rate. Before calculating, determine whether the supplied parameter is a rate or a mean and whether the event is a survival probability, an interval, a conditional residual lifetime, or a transformed payment.

Exam P often embeds exponential lifetimes in deductibles, minima, competing clocks, and conditional survival questions. The quickest reliable approach is to write the survival function first, translate every time into the parameter’s unit, and invoke memorylessness only when the condition is precisely survival beyond an elapsed time.

Recognition audit

Signals that this is the right method

A nonnegative waiting time has a constant hazard or a stated exponential mean or rate.
The requested event is naturally expressed as survival beyond a time or between two times.
The problem asks about remaining lifetime after the item has already survived to a specified age.
Independent exponential clocks are combined through their minimum or through the identity of the first clock to ring.

Before calculation

A disciplined first pass

Name the requested quantity, mark its units or time point, and identify the assumption that permits the formula. Only then insert the numerical inputs. This separates model errors from arithmetic errors and makes the answer-choice comparison auditable.

Formula core

Exponential Distribution formulas to know

These identities are starting points, not substitutes for checking support, timing, rate periods, or independence assumptions.

Density and survival

f_X(x)=\lambda e^{-\lambda x},\qquad S_X(x)=P(X>x)=e^{-\lambda x},\quad x\ge 0

If the mean is supplied, convert it first with λ = 1/E[X].

Memoryless property

P(X>s+t\mid X>s)=P(X>t)=e^{-\lambda t}

The elapsed time disappears only after conditioning on survival, not after an arbitrary event.

Moments

E[X]=\frac{1}{\lambda},\qquad \operatorname{Var}(X)=\frac{1}{\lambda^2}

Rate and scale conventions are reciprocals, a frequent source of wrong-answer choices.

Three failure modes to catch

Using the stated mean as λ instead of converting it to the reciprocal rate.
Computing a lower-tail CDF when the wording asks for survival beyond a threshold.
Applying memorylessness to a condition other than having survived to the conditioning time.

Verified practice ledger

Exam P exponential distribution problems and solutions

The list is grouped by current official syllabus area. Every link opens a step-labeled solution, verification artifact, distractor analysis, and one fully worked original variant.

General Probability

3 verified references

SOA Exam P Sample Question #84Law of Total ProbabilityVerified
SOA Exam P Sample Question #306Bayes' TheoremVerified
SOA Exam P Sample Question #605Bayes' TheoremVerified

Univariate Random Variables

73 verified references

Multivariate Random Variables

7 verified references

Continue the study loop

Keep every worked reference in one searchable manual

The complete Exam P manual includes all active sample-reference solutions, their wrong-choice analysis, and one independently written worked variant per reference. The free formula sheet is the compact companion for recall between practice sessions.

View contents and sample pagesGet the free formula sheet

Related drills

Other Exam P technique collections

Switch techniques when you can identify the current method before writing a formula and can explain each wrong-choice mechanism.

103 verified problems

Exponential Distribution practice problems