You write a stack of thank you cards for people who gave you presents for your birthday. You address all of the envelopes but before you can stuff them you are called away. A friend tying to help you see the stack of cards and stuffs them in the envelops. Unfortunately they did not realize that each card was personalized and just stick them in the envelops randomly. Assuming there were \(n\) cards and \(n\) envelops, let \(X_{n}\) be the number of cards in the correct envelope.
- Find \(\mathbf{E} (X_n)\).
- Show that the variance of \(X_n\) is the same as \(\mathbf{E} (X_n)\).
- Is there any common distribution which has the above statistics ?
- (**) Show that
\[ \lim_{n\rightarrow \infty} \mathbf{P}(X_n =m) = \frac{1}{e\, m!}\]
