Worked solution · Exam P
This Exam P sample reference tests Shifted Exponential Distribution. The density is a shift by 5 of an exponential variable with mean 5. Conditioning above 10 changes only the location of the remaining exponential lifetime, not its variance, so the conditional variance is 25, choice C.
Independent solution
Setup
Separate the fixed lower endpoint from the exponential component.
Model
The condition X at least 10 is equivalent to Y at least 5, after which memorylessness supplies a fresh exponential residual.
Compute
A deterministic shift does not affect variance, and an exponential variable with mean 5 has variance 25.
Answer
The conditional variance is 25.
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30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
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Original practice · fully worked
A shuttle delay T, in minutes, is modeled as 2 plus an exponential random variable with mean 4. Given that the delay has already exceeded 8 minutes, determine the 75th percentile of the conditional delay distribution.
Beyond minute 8, the residual delay is a fresh exponential variable with mean 4. Its 75th percentile is 4 log 4, so the total conditional percentile is 13.545, choice D.
Setup
Represent the shifted delay and translate the conditioning event to the exponential component.
Model
Memorylessness makes the conditional total equal in distribution to 8 plus a fresh exponential residual.
Compute
Set the residual CDF equal to 0.75 and add the elapsed threshold.
Answer
The conditional 75th percentile is approximately 13.545 minutes.
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