Welcome to the foundational concepts of probability and statistics crucial for actuarial exams. This module focuses on Random Variables, a cornerstone concept that bridges the gap between abstract probability and real-world phenomena. Mastering random variables is essential for modeling uncertainty in insurance, finance, and other actuarial domains.

A random variable is a variable whose value is a numerical outcome of a random phenomenon. Think of it as a function that maps outcomes from a sample space to real numbers. For example, if we flip a coin twice, the sample space is {HH, HT, TH, TT}. We could define a random variable X as the number of heads, so X(HH)=2, X(HT)=1, X(TH)=1, and X(TT)=0.

Random variables are broadly classified into two main types based on the nature of their possible values: discrete and continuous.

A discrete random variable can only take on a finite number of values or a countably infinite number of values. These are typically values that can be counted, such as the number of events occurring in a fixed interval or the number of successes in a series of trials.

A continuous random variable can take on any value within a given range or interval. These variables are typically measurements, such as height, weight, temperature, or time. For continuous random variables, the probability of the variable taking on any specific value is zero; instead, we talk about the probability of it falling within a certain range.

The probability distribution of a random variable describes the likelihood of each possible value or range of values it can take. For discrete random variables, this is the Probability Mass Function (PMF), denoted as $P(X=x)$. For continuous random variables, it's the Probability Density Function (PDF), denoted as $f(x)$, where the probability of $X$ falling within an interval $[a, b]$ is given by the integral of $f(x)$ from $a$ to $b$.

For actuarial exams, you'll encounter specific probability distributions that model common scenarios. Understanding their properties, such as their PMF/PDF, CDF, mean (expected value), and variance, is critical. Common distributions include the Bernoulli, Binomial, Poisson, Uniform, Exponential, and Normal distributions.

Mastering random variables and their distributions is a fundamental step towards understanding more complex actuarial models. Practice identifying the type of random variable in different scenarios and understanding how to calculate probabilities associated with them.