The total area under the curve must equal 1, representing the fact that the probability of some outcome occurring within the entire range is certain. \[\int_{-\infty}^{\infty}f\left(x\right)dx=1\] ...

Abstract: Alice selects an arbitrary pdf f and uses a stochastic encoder to generate a prefix-free codeword M, which is sent to Bob so that he can generate a single instance of the random variable X ~ ...

Abstract: In this chapter, we introduce the concept of a random variable and develop the procedures for characterizing random variables, including the cumulative distribution function, as well as the ...

Probability theory forms the mathematical backbone for quantifying uncertainty and random events, providing a rigorous language with which to describe both everyday phenomena and complex scientific ...

Roll a die and ask students to identify the random variable. Since a die can only take on values of 1, 2, 3, 4, 5, or 6, this is a discrete random variable. Repeat ...