Q.Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

Continuous Random Variable

A continuous random variable $X$ is described by a probability density function (p.d.f.) $f(x)$ with

Probabilities are areas under the density: $P(a\le X\le b)=\displaystyle\int_a^b f(x)\,dx$ (individual points have probability zero).

For a density that is constant on an interval (a uniform density) $f(x)=\tfrac{1}{\text{length}}$, the normalisation condition fixes any unknown constant, after which required probabilities such as $P(X<c)$ or $P(X>c)$ are obtained by integration. The mean is $E(X)=\int x f(x)\,dx$ and the variance $\int x^2 f(x)\,dx-\mu^2$.

Continuous random variables are a key part of the probability and statistics component of the Class 11-12 Mathematics curriculum, and questions on finding an unknown constant in a p.d.f. or computing P(a<X<b) are common in CBSE board papers and important-questions lists. Students searching for "continuous random variable definition, formula and examples" or "probability density function class 12 maths" will find the normalisation condition and mean/variance formulas covered here directly useful for exam preparation.