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18m
revised Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric
added 5 characters in body; edited title
20m
comment On a limit in probability theory and its convergence rate:
By the way... what happens there?
8h
comment Convergence of exponential Brownian martingale to zero almost surely
Much simpler is to show that $aW_t-\frac12a^2t\to-\infty$ almost surely, when $t\to\infty$.
10h
revised Find $P(A\Delta B)$ from $P(A)$, $P(AB)$, and $P(A^cB)$
deleted 6 characters in body; edited title; edited tags
10h
comment Find $P(A\Delta B)$ from $P(A)$, $P(AB)$, and $P(A^cB)$
@OP I just replaced every \scriptstyle \mathcal by \mathcal since \scriptstyle was unnecessary and had unfortunate formatting effects. Re the inequality in b): is your edit intended to say that $P(ABCD)$ in the LHS should actually read $P(A)P(B)P(C)P(D)$? Because then the inequality becomes obvious...
10h
comment Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$
Oops... no more "condescension" comment. So convenient.
10h
comment Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$
@DanielFischer It is plausible that futurebird has been unable to reach a sensible question for 5 days and that they responded to my tries to make them reach one by accusing me of condescension. It is plausible that I have been the only user so far to try to help the OP and that for this I have been insulted by them and chastised by a mod. So be it. Let now the multitude of users more helpful to this OP than I could ever be, enter the scene.
11h
comment Complex power series which converges absolutely on the boundary converges absolutely on a neighborhood of the boundary
Examples, examples... which examples did you try?
11h
comment Markov Chain with Memory
Sorry but it seems difficult to add more substance to answer such a vague question.
12h
comment $X_1, \dots, X_n$ are independent random variables. Suppose $M = \min(X_1, X_2, \dots, X_n)$
Sorry but, denoting by $\lambda$ the sum of every $\lambda_k$, it seems that $$E(MX_{j}:M=X_{i})=\lambda_{i}\lambda^{-2}(2\lambda^{-1}+\lambda_{j}^{-1}),$$ and $$P(M=X_i)=\lambda_{i}\lambda^{-1}.$$ The second formula was already noted by user @zoli.
13h
revised $X_1, \dots, X_n$ are independent random variables. Suppose $M = \min(X_1, X_2, \dots, X_n)$
added 57 characters in body
14h
comment Distribution of a transformed Brownian motion
$$P\left( \sup_{t \geq 0} \left(W_t - \frac12 a t\right)>x\right)=P\left(\sup_{t\geq 0}N_t>e^{ax}\right)= 1 \wedge\frac{N_0}{e^{ax}}=e^{-ax}\qquad (x>0)$$
14h
comment $X_1, \dots, X_n$ are independent random variables. Suppose $M = \min(X_1, X_2, \dots, X_n)$
Yes, $MX_j1_{M=X_i}=X_iX_j1_{M=X_i}$, not $X_iX_j$.
14h
comment probability of tail event using kolmogorov's 0-1 law
@FardadPouran "Tail event" is a standard notion of probability theory. Would you ask for the definition of a group?
14h
comment $X_1, \dots, X_n$ are independent random variables. Suppose $M = \min(X_1, X_2, \dots, X_n)$
"Now by independ(e)nce, we have $\mathbb{E}\left[MX_j:M=X_i\right] =\mathbb{E}\left[X_iX_j\right]$" Hmmm... Why exactly?
14h
comment $d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ inducing same topology on $[1,\infty)$?
Indeed you did switch. No big deal.
14h
comment $d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ inducing same topology on $[1,\infty)$?
Indeed -- and? $ $
14h
comment Find $P(A\Delta B)$ from $P(A)$, $P(AB)$, and $P(A^cB)$
About the post itself: yes please, do explain what you did to solve a). (When you will have, you might turn to b) and note that, at present, b) is just absurd. Are we missing some hypothesis?)
14h
comment $d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ inducing same topology on $[1,\infty)$?
"Clearly, we have $B_r^{d_2}(x)\subset B_r^{d_1}(x)$ for all $x\in X$" Sure about that?
15h
comment Is this backwards process a Markov chain?
@Paul Let me repeat: the backward process does not involve the inverse of the transition matrix, whether one starts from a homogenous Markov chain or not. I would be curious to see any formula characterizing the backward process and involving some entries of $P^{-1}$, when $P^{-1}$ exists.