Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 7 / b / ii

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- Solution ii

Solution

Equality can occur only at the two pairs

(0, 0)

and

(1, 1)

. Their combined probability is

e^{- p} - p (1 - e^{- p}) + p e^{- p} = (1 + 2 p) e^{- p} - p .

Taking the complementary probability gives

P (X \neq = Y) = 1 + p - (1 + 2 p) e^{- p} .

Since

1 + 2 p > 0

, the supplied exponential bound implies

P (X \neq = Y) ⩽ 1 + p - (1 + 2 p) (1 - p) = 2 p^{2} .

This is a bound on the mismatch probability for this particular coupling of the two marginal distributions; the variables are not presumed independent.

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