Continuous Probability Distributions — From Discrete to Continuous

Up to this point in the Gujarat Std-12 Statistics syllabus, the Binomial and Poisson distributions dealt with a discrete random variable — one that takes only specific, countable values (0, 1, 2, 3, …). Many real business and economic variables, however, are continuous — they can take any value within a range: the height or weight of workers, the daily closing price of a share, the marks scored by a large batch of students, the life of an electric bulb in hours, or the diameter of a machine-made component. For a continuous variable it makes no sense to ask "what is the probability that the diameter is exactly 2.500 cm?" — that probability is effectively zero. Instead, we ask for the probability that the variable falls within an interval, and this probability is represented by the area under a curve called the probability density function (p.d.f.), never by a single bar as in a discrete distribution.

Among all continuous distributions, the Normal Distribution (also called the Gaussian distribution) is the single most important one in statistics and business decision-making. It was first described by Abraham De Moivre and later developed rigorously by Carl Friedrich Gauss. Its importance comes from the fact that a very large number of naturally occurring and business-related variables — examination scores of a large class, wages of workers in a large factory, output of a production process, errors of measurement — tend to cluster symmetrically around their mean and follow this bell-shaped pattern once the number of observations is large. This chapter of the Gujarat Board (GSHSEB) Std-12 Commerce Statistics syllabus builds the normal distribution's properties, the standard normal variate, and its business applications step by step.

A continuous random variable $X$ is described by a probability density function $f(x)$ such that the total area under its curve equals 1, and the probability that $X$ lies between two values $a$ and $b$ equals the area under the curve between $x=a$ and $x=b$:

$P(a \le X \le b) = \int_a^b f(x)\,dx$

A Std-12 Commerce student is never asked to evaluate this integral directly for the normal curve — that is exactly why the standard normal table (Section 4) exists: it gives the ready-made areas so the calculation becomes simple arithmetic.