# Question about definition of 'distribution'

If a problem says to find the distribution of an RV, can I take that to mean the CDF, or take it to mean the PDF, OR is it ambiguous?

Here is a concrete example:

"If $$X_1$$ and $$X_2$$ are independent exponential random variables with respective parameters $$a$$ and $$b$$, find the distribution of $$Z = \frac{X_1}{X_2}.$$

• Intuitively speaking, the distribution of a random variable $X$ is a complete set of information needed to describe all the probability-related questions regarding $X$. So, any of CDF, PMF, or PDF (whenever appropriate) would be a valid form of answer. In general, a distribution of a random variable $X$ refers to the pushforward probability measure $$A\mapsto\mathbf{P}(X\in A).$$ When $X$ is real-valued, this is completely characterized by the CDF. If the distribution is discrete or continuous, then it is completely characterized by PMF or PDF. Nov 14, 2022 at 2:34

Commonly (or even exclusively) the term "random variable" refers to objects that take values in $$\mathbb{R}^n$$ or $$\mathbb{R}$$, where Borel sigma-algebra is defined. Then the distribution is described by a CDF (cumulative distribution function).