Discrete and Continuous Random Variables

Aim of the Experiment

The aim of this experiment is to comprehensively study and analyze the properties and behaviors of both discrete and continuous random variables through the exploration of their respective probability distributions, cumulative distribution functions (CDFs), and expected values. The experiment will specifically focus on:

1. Discrete Random Variables:

   • Defining and understanding the concept of countability in sets.
   • Investigating common discrete distributions such as Bernoulli, Geometric, Binomial, and Poisson distributions.
   • Calculating and interpreting the probability mass function (PMF), cumulative distribution function (CDF), and expected values.
   • Analyzing the independence of discrete random variables.

2. Continuous Random Variables:

   • Understanding the properties of continuous random variables, specifically focusing on the continuity of their CDFs.
   • Exploring the probability density function (PDF) and its relation to the CDF.
   • Examining common continuous distributions including Uniform, Exponential, and Normal distributions.
   • Calculating and interpreting the PDF, CDF, and expected values.
   • Evaluating the variance and standard deviation to measure the spread of these distributions.

By conducting this experiment, we aim to deepen our understanding of how different types of random variables and their distributions behave, and how their probabilistic properties can be applied to real-world scenarios. This knowledge is crucial for fields such as statistics, data science, and engineering, where probabilistic models play a fundamental role in decision-making and predictions.