Introduction to Random Variables
In probability theory, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. Unlike a regular variable in algebra that takes on a specific value, a random variable can take on different values with certain probabilities.
Random variables are mathematical functions that map outcomes from a sample space to numerical values. They provide a way to quantify and analyze random events.
Example
Consider tossing a fair coin twice. Let X be the random variable representing the number of heads obtained. Then X can take values 0, 1, or 2, with probabilities P(X = 0) = 1/4, P(X = 1) = 1/2, and P(X = 2) = 1/4.
Discrete Random Variables
Key Characteristics
- Can take on distinct, separate values
- The set of possible values can be listed (finite or countably infinite)
- Described by a Probability Mass Function (PMF)
- PMF gives the probability of each possible value
Common Examples
- Number of heads in a sequence of coin flips
- Number of defective items in a batch
- Count of customers arriving at a store
- The sum of dice rolls
Visualization of a Discrete Random Variable
Probability Mass Function (PMF) for the number of heads in 3 coin tosses
Common Discrete Distributions
Continuous Random Variables
Key Characteristics
- Can take on any value within an interval
- The set of possible values is uncountable (infinite)
- Described by a Probability Density Function (PDF)
- The probability of any exact value is zero
- Probabilities are calculated over intervals
Common Examples
- Height or weight of a randomly selected person
- Time until a component fails
- Temperature at a specific location
- Distance traveled by a particle
Visualization of a Continuous Random Variable
Probability Density Function (PDF) for a normal distribution
Common Continuous Distributions
Discrete vs Continuous: Key Differences
| Feature | Discrete Random Variables | Continuous Random Variables |
|---|---|---|
| Possible Values | Countable, distinct values | Uncountable, any value in an interval |
| Probability Description | Probability Mass Function (PMF) | Probability Density Function (PDF) |
| Probability at a Point | Can be positive (P(X = x) ≥ 0) | Always zero (P(X = x) = 0) |
| Cumulative Probability | Sum of individual probabilities | Integral of the density function |
| CDF | Step function | Smooth function |
| Expected Value Calculation | $E[X] = \sum_x x \cdot P(X = x)$ | $E[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$ |
Interactive Exploration
Select a type of random variable to visualize its properties:
Select a distribution type and click "Generate" to visualize it.
What's Next?
Now that you understand the difference between discrete and continuous random variables, you can explore their properties in more detail: