Q.Rajat is playing a game of rolling a die with his friends. According to the game rules, he will win Rs 5 for rolling an even number and for getting an odd digit on the die, he looses ₹2. If X represents the amount of money Rajat wins or loses. Show that X is a random variable and also represent it as a function on the sample space of the game play.

Random Variable: From Intuition to Definition

Imagine you're about to toss a coin three times. Before you toss, the outcome is uncertain — it could be HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT. But you're not really interested in the exact sequence. You care about something simpler: how many heads appear.

That number — the count of heads — is a random variable. It's a way of turning a random experiment into a number you can work with.

The Core Idea

A random variable is a rule that assigns a numerical value to each outcome of a random experiment. It's not "random" in the sense of being unpredictable — it's a function. The randomness comes from the experiment, not from the variable itself.

In the coin-toss example:

• Outcome HHH → value 3
• Outcome HHT → value 2
• Outcome HTH → value 2
• Outcome HTT → value 1
• and so on

The same experiment can have many different random variables. You could define:

• $X$ = number of heads
• $Y$ = 1 if the first toss is heads, 0 otherwise
• $Z$ = number of tails minus number of heads

Each is a different function on the same set of outcomes.

The Precise Definition

That's it. Three key points:

1. Domain: the sample space $S$ (all possible outcomes of the experiment)
2. Codomain: the real numbers $\mathbb{R}$
3. Rule: each outcome gets exactly one number

Why This Matters

Before the experiment, you don't know which outcome will occur, so you don't know what value $X$ will take. But you can talk about probabilities: "What's the probability that $X = 2$?" or "What's the probability that $X \geq 1$?"

This is the whole point. Random variables let you move from talking about abstract outcomes (like "HTH") to talking about concrete numbers and their probabilities. That's what makes probability a mathematical science rather than just a list of possibilities.

Two Flavors

Random variables come in two types, depending on what values they can take:

Discrete: takes only a finite or countably infinite set of values. Examples: number of heads, number of defective items in a batch, the sum of two dice.

Continuous: can take any value in an interval. Examples: height of a randomly selected student, time until a radioactive atom decays, the exact temperature at noon. …