This Exam P sample reference tests Exponential Distribution. Writing the exponential percentile through its survival probability identifies the scale parameter as 4. The median then solves exp(-m/4)=0.5, so m=4 ln 2 and choice B is correct.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe expression 5ln(5/3) linearly scales the supplied 40th-percentile value by 0.50/0.40. Exponential quantiles scale with -ln(1-p), not directly with p.
CThe value 5ln(25/12)=3.66985 has F(x)=1-exp(-x/4)=0.60047 after the scale is calibrated; it effectively targets about the 60th percentile rather than the median.
DThe value 4 is the inferred exponential mean beta. The median is beta ln 2, which is smaller than the mean.
EThe value 5 drops the logarithmic percentile conversion and reports a number from the survival ratio. With beta=4, F(5)=0.71350, not 0.50.
Original practice · fully worked
Original variant: extra repair time after escalation
A device's repair time is exponentially distributed with a median of 6 minutes. A case is escalated when its repair time passes the 75th-percentile threshold. Given that a case has just been escalated, calculate the probability that its repair takes at least 4 additional minutes.
A 0.1575
B 0.2500
C 0.5000
D 0.6300
E 0.7500
Variant answer in brief
Memorylessness removes the elapsed threshold from the conditional probability. A median of 6 gives four-minute survival 2⁽⁻⁴⁄⁶⁾=0.6300, so choice D is correct.
Setup
Setup
Convert the six-minute median into an exponential rate.
e−6λ=0.50
λ=6ln2
Model
Model
Let t denote the escalation threshold. Exponential memorylessness makes the extra four-minute survival independent of t.
Pr(T>t+4∣T>t)=Pr(T>4)
Compute
Compute
Evaluate four-minute survival using the rate obtained from the median.
Pr(T>4)=e−4ln2/6=2−2/3=0.6299605249
Answer
Answer
The conditional probability is approximately 0.6300.
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