Aim of the Experiment on Functions of Random Variables

The goal of this experiment is to explore and understand the characteristics and behavior of functions of random variables. Specifically, the experiment aims to:

1. Determine the Range and PMF of Transformed Variables:

   • Identify the range $R_Y$ for the transformed variable $Y = g(X)$.
   • Calculate the probability mass function (PMF) $P_Y(y)$ for $Y$ based on the PMF of $X$.

2. Examine Different Perspectives:

   • First Perspective: View $Y$ as a mapping from the probability space $\Omega$ to the real numbers $R_Y$.
   • Second Perspective: Consider $Y$ as the output of a system where $X$ is the input and $g(\cdot)$ is the transformation function.

3. Utilize the Law of the Unconscious Statistician (LOTUS):

   • Apply LOTUS to determine the expected value $E[g(X)]$ without needing the PMF of $Y$.
   • Validate properties like the linearity of expectation using LOTUS.

4. Analyze Transformations of Two Random Variables:

   • Study functions involving two random variables $Z = g(X, Y)$, focusing on joint distributions and the resulting marginal distributions.
   • Address problems involving the sum of two independent random variables and derive their distributions using convolution integrals.

By completing this experiment, we aim to gain a thorough understanding of how transformations of random variables impact their statistical properties, distributions, and applications in various fields.