18.44 Lecture Notes - Lecture 22: Internet Message Access Protocol, Independent And Identically Distributed Random Variables, Random Variable
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Say we have independent random variables x and y and we know their density functions f and f. Now let"s try to nd f (a) = p{x + y a} This is the integral over {(x, y): x + y a} of f(x, y) = f (x)f (y) Thus, p{x + y a} = f (x) f (y) dxdy = f (a - y)f (y) dy. Differentiating both sides give f (a) = / a f (a - y)f (y) dy = f (a - y) f (y) dy. We"re integrating over the set of x, y pairs that add up to a. It is actually one of the most important abbreviations in probability theory. Suppose that x and y are i. i. d. and uniform on [0, 1].
