Worked solution · Exam P
This Exam P sample reference tests Conditional Distributions. Conditional on a first-task duration h, the second duration is uniform on the integers from 1 through h-1 and therefore has mean h/2. Averaging over the three equally likely h values gives 11/6, choice A.
Independent solution
Setup
Let H denote the first duration and Y the second duration. The three possible values of H have equal probability.
Model
For a fixed value h, Y is discrete uniform on 1 through h-1. Its conditional mean is the midpoint of the first and last values.
Compute
Apply the law of total expectation to the three conditional means.
Answer
The expected second duration is one and five-sixths hours.
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Original practice · fully worked
A field recorder is placed with equal probability in one of three habitat classes whose species capacities are 4, 7, and 10. Conditional on capacity C=c, the number R of distinct nocturnal species detected is uniformly distributed over the integers 0, 1, ..., c. Calculate the expected value of R.
For a capacity c, the uniform count from 0 through c has mean c/2. Averaging the conditional means for capacities 4, 7, and 10 gives 3.5, so choice C is correct.
Setup
Represent the habitat capacity by C and the observed count by R.
Model
The mean of a discrete uniform distribution on the integers from 0 through c is the midpoint c/2.
Compute
Average the three conditional means with their equal habitat probabilities.
Answer
The recorder detects 3.5 distinct species on average.
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