This Exam P sample reference tests Mixture Distributions. This is a two-component exponential-mixture percentile calculation. Integrating the density and setting the survival probability to one-half gives a median of 0.904609, which rounds to choice B.
Integrate the positive-support density to obtain its cumulative distribution function. The two exponential terms carry different integration factors.
F(x)=∫0x0.4(e−0.6t+e−1.2t)dt
F(x)=1−32e−0.6x−31e−1.2x,x>0
Model
Model
At a median m, the CDF and survival probability are both one-half. Substituting y equal to the slower exponential term converts the equation to a quadratic.
32e−0.6m+31e−1.2m=21
y=e−0.6m⟹32y+31y2=21
Compute
Compute
Retain the positive quadratic root because an exponential value must lie between zero and one, then recover m with a logarithm.
2y2+4y−3=0⟹y=210−2
m=−0.61log(210−2)=0.9046093341…
Answer
Answer
The median rounds to 0.905 at the precision used by the answer list.
m≈0.905(B)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis value solves approximately f(x)=0.5, confusing the height of a density with the cumulative probability required for a median.
CThis is approximately the median of a single exponential distribution having the same mean as the mixture; a mixture is not generally replaced by that moment-matched exponential.
DSubstitution into the derived CDF gives about 0.611 rather than 0.500, so this candidate lies beyond the median.
EThis value is the mixture mean, 0.4/(0.6)²+0.4/(1.2)²=1.389, but a skewed distribution need not have equal mean and median.
Original practice · fully worked
Original variant: routed data-transfer sessions
A cloud platform routes each data-transfer session to a standard server with probability 0.25 and to an accelerated server with probability 0.75. Conditional on the route, the completion time in minutes is exponential with rate 0.4 for a standard server and 0.8 for an accelerated server. Calculate the median completion time for a randomly routed session.
A 0.405
B 0.507
C 1.014
D 1.155
E 1.733
Variant answer in brief
The unconditional completion time is a mixture, so its survival function is the route-weighted sum of the two conditional survivals. Solving the median equation gives 1.013663 minutes, selecting choice C.
Setup
Setup
Let T be the session time and condition on the randomly selected server class.
Pr(T>t)=0.25e−0.4t+0.75e−0.8t
Model
Model
For the median m, set the mixture survival probability to one-half and substitute y for the slower exponential factor.
0.25e−0.4m+0.75e−0.8m=0.5
y=e−0.4m⟹y+3y2=2
Compute
Compute
The admissible quadratic root is two-thirds. Convert that survival factor back to time.
3y2+y−2=0⟹y=32
m=−0.4log(2/3)=0.4log(3/2)=1.0136627703…
Answer
Answer
The median session completes in approximately 1.014 minutes.
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