Classical Hat Check Problem and Derangements

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Indicator Variables / Linearity of Expectation.

Your Task
At a party, $n$ guests check their hats. The hats are returned completely at random (each permutation is equally likely with probability $1/n!$). Let $X$ be the number of guests who receive their own hat. Compute the expectation $\mathbb{E}[X]$ and variance $\text{Var}(X)$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.