Upstart·Data Scientist·Recruiter / HR Screen·Intermediate

Intermediate

Jun 2026

Summary

HR screen for a data scientist role at Upstart, pretty much a probability and stats check to see if you know the basics before they bother sending you further. Nothing crazy but you better have your binomial and normal distribution stuff ready.

Questions Asked (2)

A fair coin is tossed N times independently. What is the probability of getting exactly k heads? How does the answer change if the coin has a known bias, with probability p of landing heads?

Algorithms & Data Structures

Author's notes

This is just the binomial pmf so I wasn't stressed, but I fumbled slightly explaining why the combinations term is there.

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AI HintsAI Generated

Suggested Approach

Start by identifying the distribution: for a fair coin, the number of heads follows a Binomial(N, 0.5) distribution, so the probability is C(N,k) * (0.5)^N. For a biased coin, it's Binomial(N, p) with probability C(N,k) * p^k * (1-p)^(N-k). Explain the assumptions and mention that the binomial coefficient counts the number of ways to choose which tosses are heads.

Pro tip: Emphasize that the binomial coefficient accounts for the orderings of heads and tails, and note that for large N, you might use the normal approximation or Poisson approximation when p is small, showing practical awareness.

1. Define the random variable and distribution

Let X be the number of heads in N independent tosses. For a fair coin, X ~ Binomial(N, 0.5); for a biased coin, X ~ Binomial(N, p).

2. Derive the probability mass function

For a fair coin, P(X=k) = C(N,k) * (0.5)^N. For a biased coin, P(X=k) = C(N,k) * p^k * (1-p)^(N-k). Explain that C(N,k) counts the number of sequences with exactly k heads.

3. Discuss assumptions and edge cases

Mention independence of tosses and that k must be between 0 and N. If k is outside this range, the probability is 0.

4. Mention approximations and extensions

For large N, the binomial can be approximated by a normal distribution (with mean Np and variance Np(1-p)) or by a Poisson distribution when p is small and N large. This shows deeper understanding.

Key Points to Mention

Binomial distribution and its parameters
Binomial coefficient C(N,k) = N! / (k! (N-k)!)
Independence of tosses
Probability mass function for fair and biased coins
Normal approximation for large N
Poisson approximation for rare events (small p, large N)

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.

For a normally distributed variable X with mean μ and variance σ², how do you derive P(a < X < b)? Walk through how you standardize the variable.

Algorithms & Data StructuresA/B Testing & Experimentation

Author's notes

Blanked for a second on how to phrase the standardization step cleanly.

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AI HintsAI Generated

Suggested Approach

Start by stating the goal: compute the probability that X falls between a and b. Then explain the standardization process: subtract the mean and divide by the standard deviation to convert X to a standard normal variable Z. Finally, express the probability in terms of Z and use the standard normal CDF to compute it.

Pro tip: Emphasize that standardization allows using a single table or function for all normal distributions, and mention that in practice you'd use software like Python's scipy.stats.norm.cdf rather than tables. This shows practical awareness.

1. State the problem

Clearly define that you want to find P(a < X < b) where X ~ N(μ, σ²).

2. Standardize the bounds

Transform the inequality by subtracting μ and dividing by σ: P((a-μ)/σ < (X-μ)/σ < (b-μ)/σ).

3. Introduce Z

Define Z = (X-μ)/σ, which follows a standard normal distribution N(0,1).

4. Express in terms of Z

Rewrite the probability as P((a-μ)/σ < Z < (b-μ)/σ).

5. Compute using CDF

Use the standard normal CDF Φ to compute Φ((b-μ)/σ) - Φ((a-μ)/σ).

Key Points to Mention

The standard normal distribution has mean 0 and variance 1.
Standardization formula: Z = (X - μ) / σ.
The CDF of the standard normal is denoted Φ(z).
The probability is the difference between the CDF values at the standardized bounds.
In practice, use statistical software (e.g., scipy.stats.norm.cdf) for accurate computation.
This method works for any normal distribution, regardless of μ and σ.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.