Feb 12th, 2021
Learn what it means to talk about conditional probabilities with random variables. Be able to calculate conditional expectations. Be able to use the law of total expectation.
None!
https://www.gradescope.com/courses/226051/assignments/1018506
Q: Would you mind defining obfuscate again?
A1: here, obfuscate means obscure X_i half of the time with a random coin flip. Y_i is still mathematically defined in terms of X_i, but we introduce an unpredictable fair coin flip so you’re not fully revealing the private data that might be associated with X_i.
A2: However, the statistics of X_i (expectation, for instance) can still be derived from Y_i, and if you’re only interested in the statistics, you can still get them without disclosing the private data 100% of the time.
Q: So we could also use the fact that one table sums to one accross columns and the other does across rows?
A1: absolutely.. that’s what I did :)
A2: And that’s just what Chris did too. :)
Q: do you need the parentheses in #8 or is it just there for fun?
A1: Just for fun... not needed.
Q: Is there a difference between E[X|Y] and E[X|Y=y]?
A1: nope, just two different notations.
Q: How did we do the first step?
A1: E[X|Y] is a function of Y, so the average value of that function, where each value of that function is weighted by the probability that Y = y.
Q: so we could find the expectation of S by summing over diffrent values for D_2?
A1: You certainly can. But just to clarify, do you mean computing E[S] from E[S|D_2]? That’s how I’m reading your question.
Q: Can you explain again why E[X,Y] doesn't make sense?
A1: You can only compute the expectation of a single random variable, like X, or Y, or Z = X + Y. X, Y isn’t a single random variable, because it’s not clear how X and Y are being combined.
Q: sorry I missed E[X, Y]. Is this asking E[X and Y], so a number?
A1: No, that particular one doesn’t make sense.
A2: You can only compute the expectation of a single random variable, like X, or Y, or Z = X + Y. X, Y isn’t a single random variable, because it’s not clear how X and Y are being combined.
Q: I mean we could find the expectation of S (overall) by inputing all valid numbers for D_2 (1,2,3, …) into E[E[S|D_S]]
A1: Absolutley. E[S| D_2] (which is what I think you mean) is a function of D_2, so you can now compute the weighted average of that function for all valid D_2 outcomes.
Q: how is the position of the second best engineer figured into k/i+1
A1: Technicaly, all engineers better than any in the first k are ones that stop the process. We basically want to maximize the likelihood that the best engineer isn’t among the first k engineers, and that instead is the first engineer of all better-than-first-k engineers to be interviewed once you can actually hire someone.