Compute decay, OLS, and classic probability results

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?