Do the pdf and the pmf and the cdf contain the same information? Do the pdf and the pmf and the cdf contain the same information?
For me the pdf gives the whole probability to a certain point(basically the area under the probability).
The pmf give the probability of a certain point.
The cdf give the probability under a certain point.
So to me the pdf and cdf have the same information, but the pmf does not because it gives the probability for a point x on the distribution.
 A: Where a distinction is made between probability function and density*, the pmf applies only to discrete random variables, while the pdf applies to continuous random variables.
* formal approaches can encompass both and use a single term for them
The cdf applies to any random variables, including ones that have neither a pdf nor pmf

(such as a mixed distribution - for example, consider the amount of rain in a day, or the amount of money paid in claims on a property insurance policy, either of which might be modelled by a zero-inflated continuous distribution).
The cdf for a random variable $X$ gives $P(X\leq x)$
The pmf for a discrete random variable $X$, gives $P(X=x)$.
The pdf doesn't itself give probabilities, but relative probabilities; continuous distributions don't have point probabilities. To get probabilities from pdfs you need to integrate over some interval - or take a difference of two cdf values.
It's difficult to answer the question 'do they contain the same information' because it depends on what you mean. You can go from pdf to cdf (via integration), and from pmf to cdf (via summation), and from cdf to pdf (via differentiation) and from cdf to pmf (via differencing), so when you have a pmf or a pdf, it contains the same information as the cdf (but in a sense 'encoded' in a different way).
A: The other answers point to the fact that CDFs are fundamental and must exist, whereas PDFs and PMFs are not and do not necessarily exist.
This confused and intrigued me (being a non-statistician), as I did not know how to interpret a CDF (or how it might exist) when the sample space was not ordered; think, for example, of the circle $S^1$.
It seems to me that the answer is that the fundamental function is the probability measure, which maps each (considered) subset of the sample space to a probability. Then, when they exist, the CDF, PDF and PMF arise from the probability measure.
A: PMFs are associated with discrete random variables, PDFs with continuous random variables.  For any type of random of random variable, the CDF always exists (and is unique), defined as $$F_X(x) = P\{X \leq x\}.$$
Now, depending on the support set of the random variable $X$, the density (or mass function) need not exist.  (Consider the Cantor Set and Cantor Function, the set is recursively defined by removing the center 1/3 of the unit interval, then repeating the procedure for the intervals (0, 1/3) and (2/3, 1), etc.  The function is defined as $C(x) = x$, if $x$ is in the Cantor set, and the greatest lower bound in the Cantor Set if $x$ is not a member.)  The Cantor Function is a perfectly good distribution function, if you tack on $C(x)= 0$ if $x < 0$ and $C(x) = 1$ if $1 < x$.  But this cdf has no density: $C(x)$ is continuous everywhere but its derivative is 0 almost everywhere.  No density with respect to any useful measure.
So, the answer to your question is, if a density or mass function exists, then it is a derivative of the CDF with respect to some measure.  In that sense, they carry the "the same" information.  BUT, PDFs and PMFs don't have to exist.  CDFs must exist.
