Summary

Let us summarize this chapter by revisiting the four bullet points from the beginning of the chapter.

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• Definition of a continuous random variable. Continuous random variables are measured by lengths, areas, and volumes, which are all defined by integrations. This makes them different from discrete random variables, which are measured by counts (and summations). Because of the different measures being used to define random variables, we consequently have different ways of defining expectation, variance, moments, etc., all in terms of integrations.
• Unification of discrete and continuous random variables. The unification is done by the CDF. The CDF of a discrete random variable can be written as a train of step functions. After taking the derivative, we will obtain the PDF, which is a train of impulses.
• Origin of Gaussian random variables. The origin of the Gaussian random variable lies in the fact that many observable events in engineering are sums of independent events. The summation of independent random variables is equivalent to taking convolutions of the PDFs. At the limit, they will converge to a bell-shaped function, which is the Gaussian. Gaussians are everywhere because we observe sums more often than we observe individual states.
• Transformation of random variables. Transformation of random variables is done in the CDF space. The transformation can be used to generate random numbers according to a predefined distribution. Specifically, if we want to generate random numbers according to \(F_X\), then the transformation is \(g = F_X^{-1}\).