A step-by-step walkthrough of SOA Exam P sample question 144: medical reimbursements in thousands have density proportional to x e^(−x²) on 0 to 1, and we want the mode. By the product rule and the chain rule the slope is e^(−x²)(1 − 2x²), and since the exponential is never zero the peak is where x² = 1/2, at x = √(1/2) = 0.71. We check it with the sign of 1 − 2x² (positive at 0.5, negative at 0.9) and the heights at every choice, and show why stopping at x² = 1/2 gives the tempting 0.50, plus where 1.00 (no chain rule) and 0.84 (the square root taken twice) come from. Ideal practice for the mode of a continuous distribution and the chain rule on the actuarial Exam P.
0:00 The hospital billing desk
0:17 Question 144: a peak, by the chain rule
0:29 The facts and the choices
1:03 The plan: where the slope turns
1:23 Solution: x = 0.71
1:56 Check: the sign and the heights
2:26 The trap: one step short
3:08 Takeaway
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