13.1.1. Random Variables: Discrete vs Continuous

• A random variable is a variable whose values depend on outcomes of a random phenomenon.

A random variable is essentially a variable that can take on different values based on chance or randomness. When you perform an experiment or observe something that has uncertain outcomes—like rolling a dice or drawing a card—what you observe can be expressed with a random variable. For example, when tossing a coin, the outcome can either be heads or tails. Here, the random variable can represent the number of heads you get in several tosses.

Imagine you are at a carnival and you spin a wheel with different point values on it. The point value you get after spinning the wheel depends on where the wheel stops, which is uncertain. This outcome can be represented as a random variable.

• Random variables can be: o Discrete: Takes countable values (e.g., number of heads in 5 coin tosses). o Continuous: Takes any value in a range (e.g., time, temperature, voltage, etc.).

Random variables are categorized into two types. Discrete random variables can only take specific values; these are countable and often involve whole numbers, such as the number of heads when flipping a coin multiple times. In contrast, continuous random variables can take any value within a given range and are uncountable, such as measuring someone's height or the amount of rain over a day, which can be any value within certain limits.

Consider rolling a die. The outcome, such as 1, 2, 3, 4, 5, or 6, is a discrete random variable since these values can be counted. Now think about measuring the temperature in a room. It can be 22.3 degrees or 22.4 degrees—essentially any value within a range. This measurement represents a continuous random variable.

For continuous random variables, we define a probability density function (PDF) instead of a probability mass function (PMF).

In probability theory, discrete random variables use a probability mass function (PMF) to describe the likelihood of their possible values. PMF assists in finding the probability of each outcome. Meanwhile, for continuous random variables, we use a probability density function (PDF), which provides an overall density of probabilities over intervals rather than specific values. While PMF gives specific probabilities for defined outcomes, PDF helps determine the likelihood of falling between ranges of outcomes.

Think of PMF as a bag of marbles where each marble represents a possible outcome, and you can count how many are there. Each marble has a specific color (outcome) you can count. In contrast, PDF is like a large, smooth surface where you can measure how likely you are to find an object within a certain area. The area beneath this surface represents the likelihood of different outcomes, covering the continuous spectrum of possibilities.

--