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Compute decay, OLS, and classic probability results

UpstartData Scientist · Onsite
EasyData ScientistOnsiteStatistics & Math

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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.

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
  1. 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?