Problem #451

Value of dot product
Public 12/25/17 7xp Probability 15.9%

There are two random vectors $\mathbf{a} = [a_1, a_2]$ and $\mathbf{b} = [b_1, b_2]$. Let $s$ be the dot product of these two vectors, that is $s = \mathbf{a}\cdot\mathbf{b} = a_1 b_1 + a_2 b_2$.

Assume $a_1$, $a_2$, $b_1$, $b_2$ are mutually independent random variables of the uniform distribution on the interval [0, 2]. Find the probability of value $s$ less than 5 and the expected value of $s$. Give your answer rounded to 8 digits after the decimal point.

Answer format: [probability],[expected value]
(Sample input: 0.12345678,1.23456789)

Thanks to baihacker for the idea.

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