Let $X$ be a random variable and $f(x)$ be its probability mass function. Since summation of all the probabilities equals one, it is mentioned that integration of $[f(x)\cdot dx]$ equals one.
But is it conveying the same idea ?
The integration actually gives the area beneath the curve, which need not be equal to one. Sum of probabilities equals one means that the sum of all the values (images) of $f(x)$, and not the infinitesimal areas, equals one. Right ?
Is my understanding faulty ? Please explain.
The "probability density function" for $X$ is defined by the property $$ \text{Prob}(a \le X \le b) = \int_a^b f(x)\;dx $$ whenever $a \le b$. (Of course, this does not exist in some cases, for example when $X$ is a discrete random variable.)
Then limits show us that (since all values of $X$ are real numbers) $$ 1 = \int_{-\infty}^{+\infty} f(x)\;dx $$