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As somebody used to say:

Does research. Smokes. Battles administration. Smokes. Wishes he could stop battling administration so that he could have more time to do research. Smokes some more.

The same. Except I do not smoke.

How to ask a good question?

This paragraph is for my personal use but freely available:

Welcome to Math.SE! Please, consider updating your question to include what you have tried and where you are getting stuck. That way, people on this site will know exactly what help you need.

comment Values of $\mu$ for which $S_n=e^{\sum_{i=1}^n X_i}$, is a martingale ($X_i ~ \mathcal{N}(\mu,1)$)
Still no link between $L_n$ and $Q$, hence still no real question here. (Last comment from me until a real question emerges.)
answered Probability of order statistics with numerous conditions
comment Exercise 4.4, Mathematical Analysis 2nd ed. - Apostol
A simple path is to identify $a_n$ and $b_n$ by expanding the square on the RHS, then to note that these expressions imply that $a_n-\sqrt2 b_n=(a_{n-1}-\sqrt2 b_{n-1})^2$ and finally to deduce $a_n^2-2b_n^2$ in terms of $a_{n-1}^2-2b_{n-1}^2$.
comment independence two stochastic processes
Funny: the accepted answer only proves a part of one implication (and the OP quite clearly says so), namely, that if X and Y are independent then so are U and V. Much more work is needed to solve the question, since it asks to show that if X and Y are independent Brownian motions then so are U and V, and the reverse implication.
comment Convergence of discrete random variables, show $\displaystyle\frac{S_n}{\sqrt{n}}\to0$ a.s.
Variances are not that useful to study almost sure convergence.
answered Convergence of discrete random variables, show $\displaystyle\frac{S_n}{\sqrt{n}}\to0$ a.s.
comment A question about a Markov Chain
Which similar questions can you solve? (You realize that, until now, you gave us nothing except the flat statement of your problem?)
awarded  probability-distributions
comment For an integer n ∈ N define P(n) = {primes p : p is a factor of n}.
Two users from the same class asking their homework here... Somebody should warn the TA.
comment Verifying possible solutions of the differential equation $(y')^{2}-1-y^2=0$
Are we in for a third post, once the sign problem is recognized?
answered What is the expected number of coin tosses needed to obtain a head?
comment Which of the following functions is/are solutions to the differential equation $(y'')^2-1-y^2=0$?
You "wrote the problem in a rush"--and a consequence is that several people thought about it and spent some time on it, all for nothing. Should we thank you?
comment Variance of two poisson processes
What @MichaelHardy said. Plus, I think I know what a Poisson process is, but I cannot make anything of the question... (Upvotes as mystifying as ever.)
comment Will the branching process go extinct with probability 1?
Still one more effort... Your Pk is probably $$\frac1{2^n}{n\choose k}.$$ (And then, no, extinction is not 100% sure when $n\geqslant3$.)
comment Evaluation of the series $S(\omega)=\sum\limits_{k=0}^\infty (-1)^k {\alpha \choose k}\cos(k\omega)$
Got something from an answer below?
answered Markov Chain with Memory
comment Markov Chain with Memory
Yeah, this is what VLMC (in my first comment) refers to.
comment Is it true in general that $E(1/X) = 1/E(X)$?
Sorry but the "underlying reason" seems more akin to something like $$\frac12\left(\frac1a+\frac1b\right)\ne\frac1{\frac12(a+b)}.$$
revised Transition Matrix of M/M/1 Queue
added 24 characters in body
answered Expected length of a random vector