Upstart Senior+ Data Scientist Interview Experience — Two Interviews Years Apart, Same Clash With the Interviewer

I applied to this company years ago, right after graduation, and it felt like neither I nor the phone-screen interviewer were really into each other.

Fast forward many years, and this time the HR screen and phone screen actually went well at first — but the virtual onsite took me right back to those days. I ran into another interviewer where, again, neither of us was really into the other.

That statistics round genuinely didn't go well, but the interviewer was also a piece of work. I know I didn't perform well, but it's not like they needed to call me in the middle of me working through a problem.

Question 1a
You measure temperature with one thermometer, getting results x1, x2, x3, ..., xn. The expectation of x is E(x) = T1, and the variance is σ1². What's your estimate of the temperature?

I said T1, and the interviewer said T isn't known, so I changed my answer to Σxi/n. That passed, next question.

Question 1b
There's a second thermometer, with results y1, y2, y3, ..., ym, expectation E(y) = T2, variance σ2². How would you combine the two thermometers' measurements into a single estimate? The idea is to find a linear combination of T1 and T2 that minimizes the variance.

So I started estimating the corresponding variances using x_i and y_i:

VAR(X) = Σ(xi − T1)² / (n − 1)

...and so on.

Then the interviewer cut in and asked what I was even computing, then walked me through his approach — just plugging σ1 and σ2 directly in for the coefficient, treating that as a known quantity.

Question 2a
A 2D random walk: each step has unit length, and up/down/left/right each occur with equal probability. After N steps, the coordinates are Xn, Yn. Question: what is Corr(Xn, Yn)?

Question 2b
What is Corr(|Xn|, |Yn|)?

I genuinely had no idea how to even start on 2b. Afterward I asked GPT and it didn't give me a good answer either, and running a simulation didn't get me a closed-form solution. Not sure if anyone can point me in the right direction.