Worked solution · Exam P
This Exam P sample reference tests Joint Distributions. This problem recovers two adjacent point masses from a joint cumulative table. A four-corner rectangle difference gives 0.05, which selects choice E.
Independent solution
Setup
Combine the requested adjacent masses into one rectangular event before using the cumulative table.
Model
For a joint CDF, the probability inside a rectangle is the alternating sum of its four corner values.
Compute
Insert the four cumulative probabilities at the rectangle corners.
Answer
The combined probability mass is 0.05.
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30 original multivariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
Error diagnosis
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Original practice · fully worked
A warehouse records the integer number N of delayed parcels in a dispatch. Its cumulative probabilities are F(1)=0.18, F(2)=0.41, F(3)=0.68, and F(4)=0.86. Calculate the probability that a dispatch has exactly two or exactly four delayed parcels.
Each point probability is a jump of the discrete CDF. The jumps at 2 and 4 are 0.23 and 0.18, so the requested probability is 0.41 and choice C.
Setup
For an integer-valued variable, isolate a point mass by subtracting consecutive CDF values.
Model
The two requested counts are disjoint, so their masses add.
Compute
Evaluate the two CDF jumps.
Answer
The selected counts have total probability 0.41.
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