This Exam P sample reference tests Uniform Distribution. A uniform quantile is the lower endpoint plus its percentile fraction of the interval width. Equating the two stated quantiles gives the shared endpoint parameter as 75, after which the requested midpoint is 17.5 and choice B is correct.
For a continuous uniform variable on an interval with endpoints l and u, write its r-quantile as a linear interpolation between the endpoints.
Q(r)=l+r(u−l),0<r<1
Model
Model
The first supplied interval therefore has a 40th percentile of 51. Set that value equal to the 20th percentile from the interval containing the unknown endpoint x.
25+0.40(90−25)=51
45+0.20(x−45)=51
Compute
Compute
Solve for x, substitute it into the final interval, and take that interval's midpoint.
0.20(x−45)=6⟹x=75
[10,3x]=[10,25]
Q(0.50)=10+0.50(25−10)=17.5
Answer
Answer
The requested median is 17.5.
17.5(B)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis treats the final support as [0,25] and halves the upper endpoint, giving 25/2=12.5. The actual lower endpoint is 10, so the median must average 10 and 25.
CThis reports the upper endpoint x/3=75/3=25.0 instead of the midpoint of the final uniform interval.
DThis adds the final endpoints, 10+25=35.0, but forgets the division by two in the midpoint formula (10+25)/2.
EThis is the midpoint of the intermediate interval, (45+75)/2=60.0. It answers for the wrong uniform variable.
Original practice · fully worked
Original variant: calibrate a sensor-error tail
A sensor's measurement error X is uniformly distributed on an unknown interval [L,U]. Its 10th percentile is -3 units, and its 70th percentile is 9 units. Calculate the probability that X exceeds 5 units.
A 0.250
B 0.333
C 0.500
D 0.750
E 0.900
Variant answer in brief
The two quantiles reveal a 20-unit support from -5 to 15. The threshold 5 is its midpoint, so the upper-tail probability is 0.500 and choice C is correct.
Setup
Setup
Express both stated quantiles in terms of the unknown lower endpoint and the support width W=U-L.
L+0.10W=−3
L+0.70W=9
Model
Model
Subtract the equations to determine the width, then back-substitute to recover both endpoints.
0.60W=12
W=20
Compute
Compute
The lower endpoint is -5 and the upper endpoint is 15. For a uniform variable, the tail probability is the upper-tail length divided by the full width.
L=−3−0.10(20)=−5,U=−5+20=15
Pr(X>5)=15−(−5)15−5=2010=0.500
Answer
Answer
The probability of an error above 5 units is 0.500.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
