Compute decay, OLS, and classic probability results

You will be given a set of probability and statistics questions.

1) Radioactive decay (half-life)

Each radioactive atom has a half-life of 1 day. Suppose that atoms decay independently, with decay times following an exponential distribution.

1. Starting with $n = 100$ atoms, what distribution describes the number of atoms still alive after $m = 10$ days?
2. Calculate:
   • the expected count of atoms still living after 10 days
   • the probability that at least one atom remains alive after 10 days
3. Implement a simulation that takes $m$ (days) and $n$ (number of atoms) as inputs and returns the final state of each atom after $m$ days, for example 1 for alive and 0 for decayed.

2) Population OLS coefficients

Suppose

• $y = x + \varepsilon$
• $x \sim \mathcal{N}(0,1)$ and $\varepsilon \sim \mathcal{N}(0,1)$
• $x$ and $\varepsilon$ are independent. Find the population OLS slope and intercept for:

1. Regression A: regressing $y$ on $x$ (that is, $y \sim x$)
2. Regression B: regressing $x$ on $y$ (that is, $x \sim y$)

3) Monty Hall

In the standard Monty Hall setup with three doors—one hiding a prize and two hiding goats—you choose a door, then the host opens a different door that reveals a goat and gives you an option to switch. What is the chance of winning when you switch compared with staying? Give a short justification.

4) n-sided die: time to see all faces

You repeatedly roll a fair die that has $n$ faces. What is the expected number of rolls needed to see every face at least once?

Overview: This set tests probability and statistical modeling skills: probabilistic thinking about exponential decay, the coupon-collector expectation, derivation of population OLS coefficients in simple linear models, writing a simulation, and conditional-probability reasoning as illustrated by Monty Hall.