Discrete vs. Continuous Random Variables

1. From Events to Random Variables

So far, we've talked about events like "getting heads on a coin toss" or "drawing a King from a deck." A random variable formalizes this by assigning a numerical value to every possible outcome of a random experiment. We typically denote random variables with a capital letter, like $X$.

For example, if our experiment is "flip a coin three times," our sample space $\Omega$ contains outcomes like {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. This is descriptive, but not very mathematical. We can define a random variable $X$ to be "the number of heads in three flips." Now, each outcome maps to a number:

• TTT → $X=0$
• HTT, THT, TTH → $X=1$
• HHT, HTH, THH → $X=2$
• HHH → $X=3$

Suddenly, we've transformed our abstract outcomes into a set of numbers {0, 1, 2, 3}. This allows us to ask questions like "What is the probability that $X=2$?" and use mathematical functions to describe the probabilities of all possible values of $X$.

Random variables come in two main flavors: discrete and continuous. Let's explore each.

2. Discrete Random Variables and the PMF

A discrete random variable is one that can take on a finite or countably infinite number of distinct values. Think of values you can list or count, even if the list goes on forever. There are "gaps" between the possible values.

To understand this concept better, please read the following introductory sections.

As the article explained, common examples of discrete random variables include:

• The number you get when you roll a six-sided die: {1, 2, 3, 4, 5, 6}
• The number of defective items in a sample of 20 products: {0, 1, 2, ..., 20}
• The number of emails you receive in an hour: {0, 1, 2, 3, ...} (countably infinite)

The Probability Mass Function (PMF)

For a discrete random variable, we want a function that tells us the probability of each possible value. This function is called the Probability Mass Function (PMF), denoted as $p(x)$ or $P(X=x)$. It gives the probability that the random variable $X$ is exactly equal to the value $x$.

Let's watch a video that provides an excellent visual introduction to the PMF.

As the video showed for a fair die, the PMF would be:
$P(X=x)=\frac{1}{6}\text{for }x\in \{1,2,3,4,5,6\}$
Graphically, this is a bar chart where each of the six outcomes has a bar of height 1/6.

A valid PMF must satisfy two key properties:

1. Non-negativity: $P(X=x)\ge 0$ for all possible values $x$. (Probabilities can't be negative).
2. Normalization: The sum of the probabilities for all possible values must equal 1. ${\sum }_{x}P(X=x)=1$. (Something must happen).

To solidify these properties, please review the following text.

3. Continuous Random Variables and the PDF

Now, what about quantities that aren't countable?

A continuous random variable is one that can take on any value within a given range. There are no gaps between the values; between any two possible values, there is always another possible value.

Let's start with a clear definition and some examples.

Examples of continuous random variables include:

• The exact height of a student.
• The temperature in a room.
• The time it takes to run a 100-meter race.

Here's a critical difference: for a continuous random variable, the probability of it being exactly equal to any specific value is zero. For example, what's the probability that a person's height is exactly 175.000000... cm? It's zero, because there are infinitely many possible heights around 175 cm (175.1, 175.01, 175.001, etc.).

So, we can't use a PMF. Instead of asking about the probability of an exact point, we ask about the probability of the variable falling within an interval.

The Probability Density Function (PDF)

For continuous variables, we use a Probability Density Function (PDF), denoted as $f(x)$. A PDF is a curve where:

• The total area under the curve is 1.
• The probability that $X$ falls between two values, $a$ and $b$, is the area under the curve between $a$ and $b$.

This means $P(a\le X\le b)={\int }_a^{b}f(x)dx$. Don't worry if the integration symbol is unfamiliar; for now, just think of it as a tool to find the "area under the curve." The key idea is that with a PDF, probability is area.

The value of the PDF at a point $x$, $f(x)$, is not a probability. It's a "probability density." A higher $f(x)$ value means that the random variable is more likely to be found in an interval around $x$.

Let's watch a visual explanation of the PDF to make this clear.

A valid PDF must satisfy two properties, analogous to the PMF:

1. Non-negativity: $f(x)\ge 0$ for all $x$. (The density curve cannot dip below the x-axis).
2. Normalization: The total area under the curve must be 1. ${\int }_{-\infty }^{\infty }f(x)dx=1$. (The total probability over all possible values is 1).

4. Summary and Comparison

Let's bring everything together. The fundamental difference lies in whether the variable's possible values are countable (discrete) or fill a continuous range. This determines the type of function we use to describe its probabilities.

Here's a table that summarizes the key distinctions.

This image provides another excellent visual comparison of the two concepts:

Conclusion

Congratulations! You've taken a significant step in your understanding of probability. Moving from abstract events to numerical random variables is the foundation for modeling and analyzing data.

Key Takeaways:

• A random variable assigns a number to each outcome of a random experiment.
• Discrete random variables have countable outcomes. Their probability distribution is described by a Probability Mass Function (PMF), where the value of the function at a point is the probability of that point.
• Continuous random variables have outcomes that fall within a continuous range. Their probability distribution is described by a Probability Density Function (PDF), where the area under the curve over an interval gives the probability of the variable falling in that interval.
• For a discrete variable, $P(X=x)$ can be positive. For a continuous variable, $P(X=x)=0$ for any exact value $x$.

Preview of the Next Lesson:

Now that we know how to describe the probabilities of random variables, our next step is to summarize their distributions. In the next lesson, we will learn how to calculate and interpret the expected value and variance of random variables. These two numbers will give us a powerful way to describe the "center" and "spread" of a probability distribution.