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51m
comment Dealing with Recurrence Relations of Random Variables
Obviously, for every $n\geqslant1$, $$X_n=\alpha^{n-1}Y_1+\beta\sum_{k=1}^{n-1}\alpha^{n-k-1}Y_k,$$ and the moments of $X_n$ follow. Determining the distribution is, in the general case, notoriously more difficult.
55m
comment Dealing with Recurrence Relations of Random Variables
@Paul ?? All these are already specified in the model.
58m
comment Finding the MLE for an open interval.
@Pinocchio What for?
59m
revised Moment generating function and convergent random variables
edited body
1h
revised Proof by induction: $(a+b)^n=a^n+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+…+nab^{n-1}+b^n$
edited title
1h
comment How to prove $\lim_{s \rightarrow \infty} \zeta(s) = 1$?
"Cumbersome"? If this is the proof I think, it is probably the most elementary and direct one that one can imagine.
1h
revised Calculate the discrete density of the variables of a Markov chain
edited tags
1h
comment Calculate the discrete density of the variables of a Markov chain
This is the answer for the distribution of $X_0$. Now to the distributions of $X_1$ and $X_2$...
1h
comment Defining the states when we roll one single die repeatedly
This is not about speculating on the solution the OP expected but about recognizing that each relation such as $ p_1 = \tfrac16(p_1 + p_2 + p_3 + 1 + p_5 + 0)$ uses the Markov property.
2h
comment Defining the states when we roll one single die repeatedly
"This does not use Markov chains at all" Actually this is entirely based on the (simple) Markov property after one step.
2h
revised Defining the states when we roll one single die repeatedly
deleted 39 characters in body; edited title
2h
comment Defining the states when we roll one single die repeatedly
Analytics confirm the numerical conjecture at the end.
7h
revised How could we define the factorial of a matrix?
added 12 characters in body
7h
comment How could we define the factorial of a matrix?
Better now. I corrected "Cayley" and various orthographical or syntactical oddities.
7h
comment Maximum of a sum of random variables
Seems very much so. +1 from me.
10h
comment How can I show that this function is discontinuous at the point $x=1$?
This answers the question in the title but not the question in the text.
10h
comment Maximum of a sum of random variables
You seem to be computing $E(e^{-\lambda S_n}\mathbf 1_{S_n>0})$ instead of $E(e^{-\lambda S_n^+})$. The correction to be made is easy.
11h
comment How can you model the quicksort algorithm as a martingale sequence?
Obviously $T_Q$ depends only on the choices of the pivots $(P_k)_{0\leqslant k\leqslant n-1}$, hence indeed $T_Q=E(T_Q\mid (P_k)_{0\leqslant k\leqslant n-1})=T_n$. Furthermore, for every $i$, $\mathcal F_{i-1}\subseteq\mathcal F_i$, where $\mathcal F_j=\sigma((P_k)_{0\leqslant k\leqslant j-1})$ hence, by the so-called tower property, $E(T_Q\mid \mathcal F_{i-1})=E(E(T_Q\mid \mathcal F_{i})\mid\mathcal F_{i-1})$, that is, $T_{i-1}=E(T_i\mid F_{i-1})$.
11h
comment Conditional expectation of another expectation expression.
@JimmyR. Your suggestion makes the exercise wrong, in fact, $E(E(V\mid U)\mid W)$ need not be $E(V\mid W)$ except if $W$ is $\sigma(U)$-measurable.
11h
comment Conditional expectation of another expectation expression.
When $V$ is square integrable, this says that, for some vector subspaces $J\subset K$ of $L^2(P)$ that I will let you discover, the associated orthogonal projections $\pi_J$ and $\pi_K$ are such that $\pi_J=\pi_J\circ\pi_K$.