Business Mathematics and Statistics · Ch 7 — Probability Distributions
Introduction: From Random Variables to Probability Distributions
Introduction: From Random Variables to Probability Distributions
In the previous chapter you met the idea of a random variable — a variable whose value is determined by the outcome of a chance experiment (the number of heads in three coin tosses, the number of defective pieces in a sample, the daily sales of a shop). Once you list every possible value of a random variable together with its probability, you get a probability distribution.
There are two very different ways such a list can arise:
- An empirical (observed) frequency distribution is built from actual data — you toss a coin 100 times, count how many times you got 0, 1, 2, … heads, and turn those counts into relative frequencies. It describes what did happen in one particular set of trials.
- A theoretical probability distribution is built from a mathematical law (a formula) that gives the probability of every possible value before you run a single trial, provided the situation satisfies a known set of assumptions. It describes what should happen, on average, whenever those assumptions hold.
Theoretical distributions matter for business decision-making because they let a manager compute a probability — the chance a batch has more than 2 defectives, the chance daily sales fall below a target, the chance a machine breaks down twice in a week — using only a short formula and a handful of parameters, instead of collecting fresh data every time. This is exactly the reasoning used in quality control, insurance premium-setting, inventory planning and financial risk analysis, and this syllabus draws on the same statistical principles taught across Indian commerce curricula when it comes to probability distributions questions and answers for Class 12 Business Mathematics and Statistics.
This chapter builds three such theoretical distributions, in increasing order of the kind of random variable they describe:
| Distribution | Type of variable | Typical business use |
|---|---|---|
| Binomial | Discrete, fixed number of trials | Pass/fail, defective/non-defective counts in a fixed sample |
| Poisson | Discrete, rare events over a fixed interval | Accidents/day, defects in a large batch, calls/minute |
| Normal | Continuous, bell-shaped | Sales, heights, marks, measurement errors |
Each has its own probability formula (its probability mass function, or p.m.f., for a discrete variable; its probability density function, or p.d.f., for a continuous one), its own mean and variance, and its own set of assumptions that must genuinely hold before it is honestly applied to a real business problem.
A distribution built from actually recorded data — the relative frequencies counted in one specific set of trials.
A distribution built from a mathematical formula that predicts probabilities for every possible value of a random variable, based on a set of assumptions about the underlying process, without needing to collect fresh data first.