This covers the theory of transformations of random variables: given the distribution of a random variable X and a transformation Y = g(X), the change-of-variables formula derives the density of Y from the density of X via the derivative (or, in multiple dimensions, the Jacobian determinant) of the inverse transformation, applicable when g is differentiable and monotonic (invertible). It also covers convolution, the method for finding the distribution of a sum of independent random variables by summing (discrete case) or integrating (continuous case) over the joint contributions of the summands, and the probabilistic method, a technique for proving existence of an object with a desired property by showing that a randomly chosen object has that property with positive probability, or that its expected value meets a threshold implying at least one instance achieves it. These belong to probability theory, specifically the study of distributions of transformed and combined random variables and non-constructive existence proofs.
This covers the theory of transformations of random variables: given the distribution of a random variable X and a transformation Y = g(X), the change-of-variables formula derives the density of Y fr…