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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?

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