Suppose you have a continuous random variable X with pdf and cdf . Now look at the transformation Y = X + k, with k being a real number. It's easy to see that the distribution of Y is given by and
Does this imply that Y has the same distribution as X? In other words, does a shift either to the left or to the right of the random variable preserve its distribution? It would seem obvious that the answer is "yes" (basically I'm taking the shape and moving it without distortions), but I've yet to see a reference on this yet. Any suggestions?
On the other hand, if the answer is "not necessarily", the only culprit can be a Pareto distribution, which has a funkier definition. However, I think it works even in this case (e.g. a shifted Pareto remains Pareto).