Law of the sum of two random variables. Suppose I have two independent standard normal random variables, $a_1$ and $a_2$. If I want to find the
Law of $(a_1 + a_2)$. Then does it mean that I just need to find the pdf of $(a_1 + a_2)$, with $a_1$ and $a_2$ follow standard normal distribution?
 A: The pdf of a distribution that has a pdf is one way of characterizing the distribution. The cdf is another (and works even when there is not a pdf). And there are others.
In some contexts, the answer to "Find the law" or "Find the distribution" could consist only of saying "It is normally distributed with expected value $0$ and variance $2$." or "Its distribution is $\operatorname N(0,2).$"
One can say that the density with respect to the measure $dx$ is $\displaystyle \frac1{\sqrt 2\cdot\sqrt{2\pi}} e^{-x^2/(2\cdot2)}.$
Or one can say that the distribution is
$$
\frac 1 {\sqrt 2\cdot\sqrt{2\pi}} e^{-x^2/(2\cdot 2)} \, dx.
$$
Notice that in one case above I said "the density is" and in the other I said "the distribution is", and I am careful about which is which.
One can say that the distribution is the function
$$
A \mapsto \int_A \frac 1 {\sqrt 2\cdot\sqrt{2\pi}} e^{-x^2/(2\cdot 2)} \, dx \text{ for } A\subseteq \mathbb R, 
$$
i.e. the distribution is a function whose input is a subset of $\mathbb R$ and whose output is the probability assigned to that subset.
(If one knows measure theory, one adds that $A$ must be a measurable subset.)
