# Tagged Questions

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### Markov Chain: classify states of finite Markov chain

I can easily see the states of this MC, recurrent and transient if I graph them, but how do I prove that a state is recurrent or transient. My book refers to probability to ever returning to state $j$ ...
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### $\limsup$ and probability

I cant do this: assume that $P(\limsup A_n)=1$ and $P(\liminf B_n)=1$. Prove that $P(\limsup (A_n \cap B_n))=1$. The most I get is that if $P(\liminf A_n)=1$ this is right.
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### Two dependent random variables with standard normal distribution and zero covariance

I need to find two dependent random variables with standard normal distribution, but with zero covariance. It is easy too find just two dependent random variables with such a distribution (...
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### Estimating the maximum of a Brownian motion over the unit interval

Let $\left(B_t\right)_{t \in \left[0,\infty\right)}$ be a standard Brownian motion over the probability space $\left(\Omega, \mathcal{A}, P\right)$. For each $x \in \left(0, \infty\right)$, give an ...
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### Random var. Y with pdf $f_Y(y) = 4y^3$. Show that $-2\ln (Y^4)$ ~ $X_{(2)}^2$.

Let Y be a random variable which has pdf $$f_Y(y) = \begin{cases}4y^3, & 0 < y < 1, \\ 0, &\text{elsewhere}.\end{cases}$$ Show that $-2 \ln (Y^4)$ ~ $X_{(2)}^2$. Could anyone get me ...
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### Find a asymptotic upper bound for $\sum_{n=N}^{\infty}p_{ii}^{(n)}$ for a asymetric one-dimensional simple random walk

For asymmetric one-dimensional simple random walk, that is $$P(X_n = X_{n-1} + 1) = p = 1 - P(X_n = X_{n-1} - 1)$$ for some $p \ne 1/2$, provide an asymptotic upper bound for ...
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### Help for $U = T^2$ has an $F$ distribution with 1 numerator and $v$ denominator degrees of freedom.

If $T$ has a $t$ distribution with $v$ degrees of freedom, then $U = T^2$ has an $F$ distribution with 1 numerator and $v$ denominator degrees of freedom. First, I set $$T = \frac{Z}{\sqrt(W/v)}$$, ...
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### A conditional probabilty question.

Question: $8$ identical balls are randomly distributed into $8$ boxes. Given first box and second box are not both empty, find the probability that first box is not empty? $A:=$ B1 is not ...
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### Find MLE of $\alpha$ of $f(x;\alpha)=(1+\alpha x) /2$ (stuck at derivative setup)

$X_1,...,X_n$ is an independent sample with common density: $f(x;\alpha)=(1+\alpha x) /2$ where $-1<x<1$ and $-1<\alpha <1$ I have to find the maximum likelihood estimate of $\alpha$. ...
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### Suppose that $X_1, X_2, …, X_n$ are i.i.d. random variables such that $X_1\sim N(\mu, 0.5)$ and $n = 100$.

Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d. random variables such that $X_1\sim N(\mu, 0.5)$ and $n = 100$. a) Find a maximum number $c$ such that P(X_1\le c+\mu, X_2\le c+\mu, \ldots,X_n\le ...
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### Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
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### Show that this is a stopping time

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I've been trying to put the set $\{\sigma\le t\}$ equal to a countable union and ...