1. Meaning of a Random Variable
A random variable is a rule that assigns a number to each outcome of a random experiment. The outcome itself may be a word or description, but the random variable converts it into a numerical value.
It is called “random” because the value it takes depends on the result of the experiment, which is uncertain.
1.1. Examples
- Coin toss: Let X = 1 for head, X = 0 for tail. The outcome is H or T, but X takes values 1 or 0.
- Rolling a die: Let X be the number shown. Here, the sample space is {1,2,3,4,5,6} and X takes one of these values.
- Drawing a card: Let X = 1 if a red card appears, and X = 0 otherwise.
2. Why Random Variables Are Useful
Random variables allow us to work with uncertain situations using numbers instead of descriptions. This makes calculations, formulas and probability rules easier to apply.
They help in:
- Summarising outcomes using numerical values
- Finding probabilities more easily
- Building probability distributions
- Working with averages, variances, and other statistical ideas
3. Types of Random Variables
There are two main types of random variables based on the kind of values they take.
3.1. Discrete Random Variables
A discrete random variable takes distinct, separate values. These values can be listed or counted.
Examples:
- Number on a die (1 to 6)
- Number of heads in 3 coin tosses
- Number of students present
3.2. Continuous Random Variables
A continuous random variable takes values from an entire interval. The values cannot be counted one by one, and they usually come from measurements.
Examples:
- Height of a plant (e.g., 152.3 cm)
- Time taken to finish a task
- Temperature of a day
These values can be any number within a range, not just whole numbers.
4. Examples Showing Random Variables in Action
Here are some simple examples to understand how random variables convert outcomes to numbers:
4.1. Example 1: Coin Toss
Let X = 1 if the result is a head, and X = 0 if the result is a tail.
Possible values of X: {0, 1}
4.2. Example 2: Rolling a Die
Let X be the number shown on the die.
Possible values of X: {1, 2, 3, 4, 5, 6}
4.3. Example 3: Measuring Temperature
Let X be the temperature measured at noon.
X can take any value like 28.5, 29.0, or 29.3 degrees.