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Sum of two independent uniform variables

For independent

U_{1}, U_{2} \sim Uniform (0, 1)

, the sum has triangular density

f (s) = ⎩ ⎨ ⎧ s, 2 - s, 0, 0 \leq s \leq 1, 1 \leq s \leq 2, otherwise .

Its upper tail for

1 \leq c \leq 2

is

(2 - c)^{2} /2

.

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Solution