Quick, fun, and brainy! 🧠

Solve these 2-minute probability puzzles with just a few steps — no heavy math needed.

Problem 1

An $n \times n$ matrix $X$ is filled with i.i.d standard Gaussian variables $X_{i,j}∼\mathcal{N}(0,1)$. Find $\mathbb{E}(\det{X})$ and $\mathrm{Var}(\det{X})$.

Problem 2

Points are uniformly distributed inside a square that is tilted relative to the coordinate axes (like the shape 🔷). Are the horizontal and vertical coordinates (1) linearly correlated and (2) statistically independent? Explain.

Problem 3

Find the limit

\[\lim_{n \to \infty} \frac{1}{2^n n^2} \sum_{i=1}^n {i^2 \binom{n}{i}}\]

by constructing a suitable probability distribution.

Problem 4

Suppose a series of independent random events occur, where the waiting time between consecutive events (e.g., customer arrivals) follows an exponential distribution with parameter $\lambda$. Within a fixed time interval $[0,T]$, what is the probability of observing $n$ events? Given $n$, how are the actual times of occurance $t_i~(0\le t_1 \le \cdots \le t_n \le T)$ distributed?