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Overlap between [16,24] and [18,22] is [18,22], which is fully contained. But if we model Bimodal as two pulses: from 16–18 and 20–22 (each 2 years), and uniform 18–22, overlap is 18–22 → 4 years.
But simpler: treat Bimodal as union of [16,20] and [20,24] — but assume peak interest at 18 and 22. For simplicity and alignment with discrete interpretation, the overlap of support is [18,22], so the length of intersection is 22 − 18 = <<22-18=4>>4 years.
Thus, expected overlap is 4 years.
A robotics engineer builds a hierarchical robot system with one command unit, 4 sensor clusters (all same, but placed differently), and 3 task modules (distinct functions: navigation, analysis, repair). Each sensor cluster can be assigned to any of 4 mounting positions, but due to space constraints, only 2 clusters can be placed per unit. How many distinct robot setups can be created?
First, choose 2 sensor clusters out of 4: C(4,2) = 6.
For each pair, assign the 2 chosen clusters to 4 distinct mounting positions with no two in the same spot: permutation P(4,2) = 4 × 3 = 12.
Command unit: fixed (1 choice).
Task modules: 3 distinct modules, assign one to each of 3 unique functions — 3! = 6 ways.
Total setups: 6 (sensor clusters) × 12 (placement) × 6 (task modules) = 6 × 12 × 6 = <<6*12*6=432>>432.
#### 432**Question:** A linguist is studying the evolution of language through a model where the frequency of a certain linguistic feature in a population is given by the function \( f(t) = 3t^2 - 2t + 5 \). Find the time \( t \) at which the frequency is 20.