You are asked several probability/statistics questions. ## 1) Radioactive decay (half-life) A radioactive atom has a half-life of **1 day**. Assume each atom decays independently and decay follows an exponential distribution. 1. If you start with **n = 100** atoms, what is the probability distribution of the number of atoms still alive after **m = 10** days? 2. Compute: - the **expected** number of atoms still alive after 10 days - the probability that **at least one** atom is still alive after 10 days 3. Write a simulation: input **m** (days) and **n** (number of atoms) and output the **final state** of each atom (e.g., 1 = alive, 0 = decayed) after m days. ## 2) Population OLS coefficients Let - \(y = x + \varepsilon\) - \(x \sim \mathcal{N}(0,1)\), \(\varepsilon \sim \mathcal{N}(0,1)\) - \(x\) and \(\varepsilon\) are independent. Compute the population OLS slope (and intercept) for: 1. Regression A: \(y\) on \(x\) (i.e., \(y \sim x\)) 2. Regression B: \(x\) on \(y\) (i.e., \(x \sim y\)) ## 3) Monty Hall In the classic Monty Hall problem with 3 doors (1 prize, 2 goats), after you pick a door the host opens a different door showing a goat and offers you the chance to switch. What is the probability of winning if you **switch** vs if you **stay**? Briefly justify. ## 4) n-sided die: time to see all faces You roll a fair die with **n** faces repeatedly. What is the **expected** number of rolls required to observe **all n faces at least once**?