Compute decay, OLS, and classic probability results
Company: Upstart
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Onsite
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**?
Quick Answer: This question evaluates probability and statistical modeling skills—probabilistic reasoning for exponential decay and the coupon-collector expectation, derivation of population OLS coefficients for simple linear models, simulation implementation, and conditional probability reasoning exemplified by the Monty Hall scenario.