I ask A LOT of questions when I learn something. (Huge thanks to all of my instructors for not getting annoyed ^ ^)I’m going to keep track of the questions I’ve asked that I may want to revisit in the future.
- [Probability] Poisson distribution & normal distribution?
- Question: We use Poisson distribution to approximate Binomial distribution, which means it’s approximately a summation of Bernoulli random variables. Since the underlying assumption for Poisson distribution is n goes go infinity, why wouldn’t Poisson distribution look like normal distribution according to CLT?
- Answer: (Prof. Adhikari; rephrased) If we fix , the sum of Bernoulli random variables will be approximately normally distribution. However, for Poisson approximation, goes smaller when goes larger.
- [Probability] Why do we use square brackets for expectations?
- Question: I notice that we write E[X] or E(X) but I’ve never seen Var[X]. Why do we sometimes use square brackets for expectations?
- Answer: (Dibya) In algebra, linear functionals are denoted by f[x] instead of f(x). A functional of a vector space is a linear map f:V→R. On the vector space of random variables, the expectation operator is a linear functional, and so often denoted as E[X]. The variance is not a linear functional, and so not written using square brackets.
- [Probability] A urn contains white balls and black balls. After a ball is drawn, it is replaced along with more balls of the same color. Show that (second ball drawn is white) = .
- Question: What is the intuition behind (second ball drawn is white) doesn’t depend on ?
- Answer: (Prof. Ibser) Consider the extreme cases: d=0 and d very large. If d=0, then the draws are independent and P(white on second draw) is same as P(white on first). If d is very large, P(white on second|white on first) is close to 1, and P(white on second|black on first) is close to 0, so P(W 2nd) = P(W 1st)P(W 2nd|W 1st) + P(B 1st)P(W 2nd|B 1st) which gets us back to P(W 1st). Hopefully thinking about the extreme cases helps with the intuition.
- [Propositional logic] is not equivalent to
- Question: Since P(x) doesn’t depend on y, what’s special about this expression that alters the expression when we move ?
- Answer: (Hung) The main difference is between “If there exists y such that …, then …” versus “There exists y such that if …, then …” . The first statement is true when that y does not exist, but the second statement would be false.
- [Misc. ] Markov Chain Monte Carlo
- Question: Where does this name come from? Why Monte Carlo?
- Answer: (Jason) Monte Carlo algorithms use random sampling to compute deterministic quantities such as integrals or probability distributions.
In the 1940s, when Stanislaw Ulam and John von Neumann were testing nuclear bombs, Ulam needed a method to estimate how much energy neutrons would give off. A deterministic model would have been too complex, so Ulam came up with the modern version of the Monte Carlo Method. Nicholas Metropolis came up with the codename for this method, naming it after the Monte Carlo Casino in Monaco.