Probability theory distinguishes between two types of random variables: discrete and continuous. The nature of the variable influences the way we calculate probabilities.

### Discrete Variables:

**Definition:**

- Discrete random variables can take on a finite or countably infinite number of distinct values.
- Examples include the number of heads in multiple coin tosses, the count of defective items in a batch, or the number of students in a classroom.

**Probability Mass Function (PMF):**

- For discrete variables, the probability mass function (PMF) is used to describe the probability distribution.
- The PMF gives the probability of each possible outcome.
- Symbolically, ( P(X = x) ) represents the probability that the discrete random variable ( X ) takes on the specific value ( x ).

**Properties:**

- ( P(X = x) \geq 0 ) for all ( x ) in the range of ( X ).
- ( \sum P(X = x) = 1 ) over all possible values of ( x ).

### Continuous Variables:

**Definition:**

- Continuous random variables can take on an uncountably infinite number of possible values within a range.
- Examples include height, weight, time, and temperature.

**Probability Density Function (PDF):**

- For continuous variables, the probability density function (PDF) describes the likelihood of the variable falling within a specific range.
- The PDF is represented by ( f(x) ), and the probability that the variable falls within an interval ([a, b]) is given by the integral ( \int_{a}^{b} f(x) \, dx ).
- Unlike the PMF, the PDF does not directly give the probability of a specific value but provides the relative likelihood of values within a range.

**Properties:**

- ( f(x) \geq 0 ) for all ( x ) in the range of the variable.
- ( \int_{-\infty}^{\infty} f(x) \, dx = 1 ).

### Example:

**Discrete Variable:**

Consider rolling a fair six-sided die. The probability mass function (PMF) is:

[ P(X = x) = \frac{1}{6} \text{ for } x = 1, 2, 3, 4, 5, 6 ]

**Continuous Variable:**

Consider a continuous variable representing the height of individuals. The probability density function (PDF) might follow a normal distribution. The probability of an individual having a height between 160 cm and 170 cm is given by:

[ \int_{160}^{170} f(x) \, dx ]

In practice, understanding whether a variable is discrete or continuous helps in choosing the appropriate probability model (PMF or PDF) and applying the relevant mathematical tools for analysis. Discrete variables are associated with probabilities assigned to specific values, while continuous variables involve probabilities over intervals.