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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 $\pi_J=\pi_J\circ\pi_K$.