Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 1 / c

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- Solution c

Solution

Suppose

c > 1

and

E [T_{c}] < \infty

. Part (a) would give

E [S_{T_{c}}^{2}] = E [T_{c}] .

But

T_{c} \geq 1

and its defining strict inequality gives

S_{T_{c}}^{2} > c^{2} T_{c}

almost surely. Taking expectations would yield

E [T_{c}] > c^{2} E [T_{c}],

which is impossible because

c^{2} > 1

. Part (b) shows that

T_{c}

is nevertheless finite almost surely. Consequently

E [T_{c}] = \infty.

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