Q.Distinguish between a discrete random variable and a continuous random variable. Give one business-related example of each.

A discrete random variable can only take a countable list of separate values (like 0, 1, 2, ...), while a continuous random variable can take any value within a range, with infinitely many possible values between any two points.

A random variable is a function that assigns a real number to every outcome of a random experiment. Based on the values it can take, a random variable is classified as discrete or continuous.

Discrete random variable. $X$ is discrete if it can take only a finite, or countably infinite, list of values $x_1, x_2, x_3, \ldots$, each with its own definite probability $p_i = P(X = x_i)$, where $p_i \geq 0$ and $\sum p_i = 1$. Between two consecutive possible values (say $1$ and $2$), $X$ cannot take any value in between.

Business example: Let $X$ = the number of defective items found when a batch of 10 items is inspected. $X$ can only be $0, 1, 2, \ldots, 10$ — a finite, countable list — so $X$ is discrete.

Continuous random variable. $X$ is continuous if it can take any value within some interval, so that there are infinitely many possible values between any two points, and no single value has a positive probability (instead, $X$ is described by a probability density function, and probabilities correspond to areas under its curve over a range).

Business example: Let $X$ = the weight (in kilograms) of a randomly selected packet coming off a filling line. $X$ could be $0.998$ kg, $0.9983$ kg, $0.99831$ kg, and so on — there is no smallest gap between possible values — so $X$ is continuous.

Key contrast: a discrete random variable is described by listing probabilities against each separate value (a probability mass function); a continuous random variable is described by a smooth curve (a probability density function) where probability is read off as area, not height.

A common mistake is to think 'discrete' means 'small number of values' and 'continuous' means 'large number of values' — the real test is whether the values can be listed one after another (countable) or fill up an entire interval (uncountable). Another common error is treating money amounts as automatically continuous; in many syllabus problems profit/loss amounts are still treated as a discrete random variable because only a specific finite list of outcome amounts is given.