# Expected number of parts of a uniformly selected partition of $n$

I have a very basic question on partition theory, which I feel should be very well known. Suppose that you fix a natural number $$n$$ and select a random partition $$P$$ of $$n$$ by choosing uniformly from the set of all partitions of $$n$$ (so that each partition has probability $$\frac{1}{p(n)}$$ of being selected, where $$p(n)$$ stands for the number of partitions of $$n$$). My question is that, what is the best known estimate/approximation of the expected number of parts of $$P$$ and the length of the largest part of $$P$$?

Since the there is an inherent symmetry between the number of parts and the length of the largest part of a partition (which is more clear if one views the Young's diagram), and since the area of the Young's diagram is $$n$$, I suspect that the expectation is roughly of the order of $$\sqrt{n}$$, although this is a heuristic guess. Can anyone give me the best known estimate?