Random Variables — Discrete and Continuous

In the earlier study of probability we assigned probabilities to events — subsets of a sample space. Very often, however, what we really care about is a number attached to each outcome: the number of heads when three coins are tossed, the number of defective bulbs in a carton, the daily demand for a product, the lifetime of an electric lamp. A random variable is exactly this idea made precise.

For example, if three coins are tossed and $X$ = "number of heads", then to the outcome $HHT$ the variable assigns $X = 2$, to $TTT$ it assigns $X = 0$, and so on. The possible values of $X$ here are $0, 1, 2, 3$.

Random variables are of two kinds, and the whole chapter divides along this line:

• A discrete random variable takes only a countable set of separate (isolated) values — usually whole numbers arising from counting. Examples: number of heads in tosses of coins, number of accidents in a day, number of customers in a queue. Between two neighbouring values (say $2$ and $3$) it takes no value.
• A continuous random variable can take any value in an interval of the real line — the values it may take cannot be listed one by one. Examples: the height of a student, the exact time a bulb lasts, the weight of a packet. Such quantities arise from measuring rather than counting.

The probability behaviour of a discrete variable is described by a probability mass function (Section 2); that of a continuous variable by a probability density function (Section 3). Both are collected under the single name probability distribution — a complete description of how the total probability $1$ is shared among the values the variable can take.

This treatment develops probability distributions from the same foundational principles used in standard senior-secondary mathematics and statistics, on which the Maharashtra Std XII Mathematics and Statistics (Commerce) syllabus is based.