Why is there a preference to use the cumulative distribution function to characterise a random variable instead of the probability density function?

Solution 1:

Every random variable has a CDF. Not every random variable has a pdf (for instance, discrete or mixtures of discrete and continuous distributions). For instance, in the Kolmogorov-Smirnov test, you are comparing an empirical CDF, which is discrete, to a potentially continuous CDF. In addition, convergence in distribution is defined using CDFs.

Solution 2:

The CDF of a random variable completely determines the distribution of a random variable.

The PDF of a random variable completely determines the distribution of a random variable.

So uptil here no differences, but they arise if we look at the converse of these statements.

The distribution of a random variable is determining for its CDF, or - in other words - there is only one CDF.

This cannot be said about PDF.

Observe for instance that for uniform distribution on $[0,1]$ we can use several PDF's. The most common one is the indicator function of $[0,1]$ but another is the indicator function of $[0,1]\cap\mathbb Q^{\complement}$.

The CDF of a random variable always exists.

The PDF of a random variable does not always exist.

Solution 3:

You pretty much always want to calculate Pr(a < x < b), which is a simple subtraction if you have a CDF. If you're using a PDF, you have to integrate, which is a very complex operation.

Solution 4:

The values of a CDF are probabilities, and thus easy to directly reason about. On the other hand the values of a PDF (when it exists) have no direct intuitive meaning but are instead characterized less directly by the property that when you integrate over a set you get the probability of being in that set. That is a little harder to reason about. CDFs are thus more intuitive than PDFs.

On the other hand, in the discrete case both CDFs and PMFs can be directly interpreted as probabilities, with the PMF giving the probability of a simpler event. In such situations the PMF is often both easier to work with and perhaps more intuitive.