 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability/9/a/ii/solution

X

has finite mean

μ

σ^{2}

, then for

a > 0

P (∣ X - μ ∣ \geq a) \leq \frac{σ ^{2}}{a ^{2}} .

Y = (X - μ)^{2}

with threshold

a^{2}

. Since

E [Y] = σ^{2}

and

{∣ X - μ ∣ \geq a} = {Y \geq a^{2}}

, the desired statement follows immediately.

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