Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 207 / 1 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

Because the measurement can cross the threshold in both directions while the model declares dementia irreversible, one needs the diagnostic rule's sensitivity

P (X < 10 ∣ true dementia)

and specificity

P (X \geq 10 ∣ no true dementia) .

These quantify false negative and false positive state classifications and permit a hidden-state or misclassification model instead of treating every threshold crossing as the true irreversible onset.

ii

Solution

If

T

is future time to dementia from age 50, its survival function is

P (T > t) = {e^{- t /10}, e^{- 1} e^{- (t - 10) /5}, 0 \leq t \leq 10, t > 10.

The tail-sum formula for expectation gives the expected future time alive without dementia:

E T = \int_{0}^{10} e^{- t /10} d t + \int_{10}^{\infty} e^{- 1} e^{- (t - 10) /5} d t = 10 (1 - e^{- 1}) + 5 e^{- 1} = 10 - 5 e^{- 1} .

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