Q.Verify which of the following is p.d.f. of r.v. X :

For a continuous random variable $X$ defined on an interval $S = (a, b)$, a probability density function $f(x)$ is a non-negative, integrable function satisfying three conditions: $f(x) \ge 0$ for every $x$ in $S$; the total area under the curve $y = f(x)$ over $S$ equals 1, i.e. $\int_S f(x)\,dx = 1$; and for any interval $A$ inside $S$, the probability that $X$ falls in $A$ is the area under $f$ over that interval, $P[X \in A] = \int_A f(x)\,dx$. Crucially, $f(x)$ itself is not a probability (it can exceed 1) — it is only areas under the curve, obtained by integrating $f$, that give probabilities. A direct consequence is that for a continuous random variable, the probability of $X$ equalling any single specific value is always 0, since the "area" over a single point is zero; only probabilities over an interval are meaningful.