 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/m5/b/i/solution

P (∣ Y - μ ∣ \geq r) \leq σ^{2} / r^{2}

for

r > 0

when

Y

μ

and finite variance

σ^{2}

. Here each summand has mean zero and variance one, so independence gives

E S_{n} = 0

and

Var S_{n} = n

. Taking

r = an

gives

P (∣ S_{n} ∣ \geq an) \leq min (1, \frac{1}{a ^{2} n}) .

 New to topics? Read the docs here!