Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-...

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1answer
53 views

How to interpret the covariance matrix of Brownian motion

I'm reading Bernt Oksendal's "Stochastic Differential Equations". It says, Brownian motion $B_t$ is Gaussian Process, i.e. for all $0 \leq t1 \leq \cdots \leq t_k$ the random variable $Z = (B_{t_1},...
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1answer
34 views

What is the difference between algebra and $\sigma$-algebra generated in a finite space?

For a finite set $\Omega$ with (obviously finite) class $C$. Does the $\sigma$-algebra generated by $C$ $=$ the algebra generated by $C$? I think it does. But I'm not sure I understand if there is a ...
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1answer
71 views

Distribution of Brownian motion before stoping time.

Let $B_{t}$ be a standard Brownian motion. Stopping time $\tau_{a} = \inf \{t \ge 0: |B_{t}| = a\}$. How to find $E[B_{\frac{\tau_{a}}{2}}]$? Or where is it possible to read about it? Thanks in ...
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0answers
33 views

Marginalization and integration of pdfs

Suppose that we have to r.v. $X_0$ and $X_1$ and $Y$. Is the following integration correct \begin{align*} &\int\int \int x_0 x_1 f_{X_0|Y}(x_0|y)f_{X_1|Y}(x_1|y) f_{Y}(y) dx_0 dx_1 dy\\ &=\...
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1answer
42 views

Existence of a random variable given a cdf

For every real function F which can be a CDF (so has the properties that $F(+\infty)=1$, $F(-\infty)=0$, and F is non-decreasing and right continuous), does there exist a random variable on a ...
2
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1answer
40 views

Convergence of a product of sequences convergent in mean when one of them is bounded

Suppose $X_n\to X$ in $L^1$ and $V_n\to V$ in $L^1$ and $(V_n)$ is a bounded sequence. I'm trying to show that then $\mathbb{E}X_nV_n\to \mathbb{E}XV$. One has for all $N\in\mathbb{N}$ $$|\mathbb{E}...
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1answer
169 views

Number of ways to set 3 queens to attack each other [closed]

We play chess and want to set 3 queens to attack each other. How many ways we can do it? I know to solve this problem when I have 2 queens. I see the chess board as 4 squares, from an outer square (...
2
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1answer
110 views

How to show that A, B and C are not independent?

Give an example of 3 events A,B,C which are pairwise independent but not independent. Hint: find an example where whether C occurs is completely determined if we know whether A occurred and whether B ...
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1answer
24 views

Two inequalities involving distribuants and expected value

I have to show that if $X \ge 0$ then $$\sum_{n=1}^\infty P(X \ge n) \le E[X] \le 1 + \sum_{n=1}^\infty P(X \ge n).$$ I know that if the random variable $X$ takes only values in $\mathbb N$, we have: ...
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1answer
41 views

If $P(A) = 0$, then can we conclude that $Q(A) = 0 $ ??

Suppose $( \Omega, \mathscr{A}, P ) $ is probability space. $X$ is r.v. such that $X \geq 0 $ almost surely and $\mathbb{E} \{ X \} = 1 $. Suppose $\exists A \in \mathscr{A}$ s.t. $P(A) = 0$, is it ...
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1answer
29 views

Let X be a non-negative continuous r.v. with pdf f(x)

Let $G(t)=\int_t^\infty$$f(x)dx$ Show that $E[X^2] = 2\int_0^\infty$$tG(t)dt$ I have not taken a course in probability in years and remember a theorem where X has a density function $f$ and ...
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2answers
42 views

What range of values of lambda does the mean of $Y$ converge? What is the mean in that case?

$X$ is an exponential RV with parameter lambda and $Y = e^x$. So, I found the density of $Y$ to be $\lambda y^{-\lambda}e^{-y}$. Then to find the range of $\lambda$ where the mean converges, do we ...
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0answers
38 views

$L_p$ inequality for Conditional Expectations with Mismatched Distributions

Suppose, we define $E_F[X]= \int X dF$ and $E_P[X] =\int X dP$. We know that there exits the follow $L_p, \ p \ge 1$ inequality \begin{align*} E_P |E_P(X| \mathcal{B})|^p \le E_P[|X|^p] \end{align*} ...
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1answer
110 views

Express the distribution function Y = max{X, 0} in terms of the distribution function of X.

This is just asking for a general case, with general distribution of X. I treated it similar to a minimum problem and said that F(y) is P(x >= 0) for x > 0, P(x = 0) for x = 0 and 0 if 0 > x. Is this ...
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1answer
20 views

For what c is $F(x)=c\int_{-\infty}^x e^{-|x|} dx$ a distribution function?

I know the properties of distribution functions, that as $x$ approaches negative infinite $F(x)= 0$ and as $x$ approaches positive infinite then $F(x) = 1$. We can't use the first option to find $c$, ...
2
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1answer
41 views

An independent sequence of square-integrable random variables with convergent sum of variances converges stochastically

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of independent and square-integrable random variables with $\operatorname{E}\left[X_i\right]=0$ and $$\sum_{i\in\mathbb{N}}\operatorname{Var}\left[X_i\right]&...
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0answers
19 views

When does $\frac{{E[{X_1}]}}{{E[{X_2}]}} > 1$ imply $\frac{{E[X_1^2]}}{{E[X_2^2]}} > \frac{{E[{X_1}]}}{{E[{X_2}]}}$?

There are two random variables ${X_1},{X_2} \in [a,b]$ where $a > 0$. When does $\frac{{E[{X_1}]}}{{E[{X_2}]}} > 1$ imply $\frac{{E[X_1^2]}}{{E[X_2^2]}} > \frac{{E[{X_1}]}}{{E[{X_2}]}}$ ?
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0answers
100 views

Conditional expectation of a bounded random variable

Suppose we have a bounded random variable $X:(\Omega,\mathcal{F})\rightarrow(\mathbb{R},\mathcal{B})$ and some other random variable $Y:(\Omega,\mathcal{F})\rightarrow(\mathbb{R},\mathcal{B})$ . ...
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0answers
55 views

Distribution of Brownian motion at a stopping time

Let $X,Y$ be independent Brownian motions, and let $$ T_a = \inf\{t\geq 0 : Y_t = a\} $$ for some $a > 0$. My question is how can I find the distribution of $X_{T_a}$? The hints I have are to use ...
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0answers
60 views

probability measure on space of sequences

Let $\Omega=\{0,1\}^\infty$. For some $n$, let $B\subset \{0,1\}^n$. I have seen these two statements which make me confused little bit. (1) If $A\subset \Omega$, $A=B\times \{0,1\}^\infty$, and ...
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2answers
43 views

If $X_1,X_2…$ are independent, uniform random variables do there exist an infinite amount of $Y_n = X_nX_{n+1} < \frac{1}{8n}$?

If $X_1,X_2...$ are independent, uniform random variables on the interval $[0,1]$, do there exist an infinite amount of $Y_n = X_nX_{n+1} < \frac{1}{8n}$? I want to use the Borel-Cantelli lemma, ...
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0answers
143 views

Probability on entering direction of a simple random walk

Let $X(n)$ be a simple random walk on $\Bbb{Z}^2$. Also we define $S_{R} = \inf\{n > 0 : X(n) \notin [-R, R]^2 \} $ : the exit time of the square $[-R, R]^2$, $T_{v} = \inf\{n > 0 : X(n) = v\}$...
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1answer
43 views

How do I combine non-overlapping distributions?

I have two distributions, one only positive and one only negative. I'm trying to figure out how to combine them mathematically into a combined distribution. For example, I have two log normal ...
3
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0answers
22 views

If $X_1,X_2$ are independent beta then showing $\sqrt{X_1X_2}$ is beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities $\beta(n_1,n_2)$...
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2answers
39 views

Why does it hold $\operatorname{E}[Y\mid\mathcal{F}]=\operatorname{E}[Y\mid Y]=Y$, if $Y$ is $\mathcal{F}$-measurable?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathcal{F}\subseteq\mathcal{A}$ be a $\sigma$-algebra on $\Omega$ $Y\in\mathcal{L}^1(\Omega,\mathcal{A},\operatorname{P})$ be ...
0
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1answer
89 views

If $\int f\;d\mu=\int g\;d\mu$, then $f\equiv g$ almost everywhere

I'm wondering whether or not the following statement is true: Let $(\Omega,\mathcal{A},\mu)$ be a measure space $f,g:\Omega\to\overline{\mathbb{R}}$ be measurable with respect to $\mathcal{A}$ and $$...
2
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1answer
168 views

Expected value complex random variable

I want to check that if $X: \Omega \to \mathbb{C}$ is a random variable, then the inequality $| \mathbb{E} X| \le \mathbb{E} |X|$ also holds like in the real case. We can write $$X = \Re X + i \cdot ...
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2answers
35 views

Elementary Probability problem

I think this is true, but i can't seem to prove it: Let A,B be events in $\Omega$ and $C_i$ be a partition of $\Omega$. $$P(A | B) = \sum_i P(A | B \cap C_i)$$
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1answer
56 views

Divergence in probability

The sequence $(X_n)$ is said to diverge to $+\infty$ in probability if $\mathbb{P}\{X_n>b\}\to 1$ as $n\to\infty$ for every $b\in\mathbb{R}_+$. If $(X_n)$ diverges to $+\infty$ in probability and $...
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1answer
106 views

Interpretation of measure theory question

I am struggling with the following question: Let $\Omega = \{1,2,3,4\}$, $\mathscr E = \{\{1\}\{1,2\}\}$ (a) Find $\sigma (\mathscr E)$ (sigma algebra) (b) By inventing an analog of a pmf or cdf, ...
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1answer
29 views

Borel measurable function, Borel set, probability

Show that for two independent random variables $X: \Omega \to \mathbb{R}^m, Y: \Omega \to \mathbb{R}^n$, Borel measurable function $g: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}$ and a Borel set $...
2
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1answer
79 views

How to prove any graph $G=(V,E)$ has a $k$-colorable subgraph with $\geq (1-1/k)|E|$ edges?

I'm trying to find a way to prove that, for each $k\in \mathbb{N}$ and each simple undirected graph $G=(V,E)$, $G$ has a subgraph $H=(V',E')$ with chromatic number at most $k$ such that $|E^\prime|\ge ...
2
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1answer
43 views

Showing a set is a Borel set and its probability is $0$

Suppose $P(X) = \int_{-\infty}^{\infty} 1_X(x) f(x) \,dx $ for $f(x) \geq 0 $ for all $x$ such that $\int_{- \infty}^{\infty} f(x)\,dx = 1 $. Suppose $A = \{ x_0 \} $. Then $A$ is a Borel set and $P(A)...
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0answers
33 views

Multivariate Gaussian Problem

The problem is like this: Conditional on $M=m, Y_1, Y_2, ..., Y_n$ is a random sample from the $N(m,\sigma^2)$ distribution. Find the unconditional joint distribution of $Y_1, Y_2, ..., Y_n$ when $M$ ...
3
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1answer
121 views

Picking Random Elements from Set

Let $S$ be a set consisting of $6$ positiver integers and $8$ negative integers. Choose a 4-element subset of $S$ uniformly at random, and multiply the elements in this subset. Denote the product by $...
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0answers
55 views

Convergence of derivative in $L^1(\mathbb R)$

Let $F(x)$ be a distribution function over $\mathbb{R}$ with positive derivative at origin $f(0)$. Let $Q$ be a measure on $\mathcal{B}(\mathbb{R})$. Can we have following results under some ...
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0answers
100 views

Projective family of probability spaces

I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})$. Let our indexing set be a poset $T$. The conditions $f_{tt}=1_{S_t}$ $f_{us}=f_{ts}\...
3
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1answer
85 views

the infinite sum of symmetric random variables is also symmetric

Definition. Let $(\Omega, {\mathcal F}, \mathbb{P})$ be a probability space and $X$ a random variable in $\Omega$. $X$ is said to be ${\mathbf symmetric}$ (about $0$) if $X$ and $-X$ are equal in law....
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1answer
41 views

Integrable variable $X$ , integral of the expected value of $|X|$ over a small set $A$

Let $(\Omega, \Sigma, P)$ be our probability space. Prove that a random variable $X$ is integrable, that is $\mathbb{E}(X) < \infty$ $\iff$ $$\forall \varepsilon >0 \exists \delta>0 : \...
3
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1answer
234 views

If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?

The random variables take values in $\mathbb{R}^d$. I have tried to prove this using characteristic functions. Let $\hat{\mu}_{X_n},\hat{\mu}_{Y_n},\hat{\mu}_{Z_n}$ be the characteristic functions of ...
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1answer
51 views

Expected value, borel measurable function, dependent variables

We are given two random vectors: $X: \Omega \to \mathbb{R}^m$ and $Y: \Omega \to \mathbb{R}^n$ - not necessarily independent, and a Borel measurable function: $g: \mathbb{R}^m \times \mathbb{R}^n \to \...
1
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1answer
62 views

Let $X_n$ be the $n$-th partial sum of i.i.d. centralized rv and $\mathcal{F}_m:=\sigma(X_n,n\le m)$, then $\text{E}[X_n\mid\mathcal{F}_m]=X_m$

Let $(\Omega,\mathcal{F},\text{P})$ be a probability space $\left(Y_i\right)_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables $(\Omega,\mathcal{F})\to\left(\mathbb{R},\mathcal{B}\left(\...
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2answers
36 views

Something about Markov chains

We check $P(X_{n+1}\in B|\mathcal{F}_n)=P(X_{n+1}\in B|X_n)$ when we want to prove $X_n,n=1,2,\dots$ is a Markov chain. Through this equation it seems that $X_n$ is a Markov chain if $X_{n+1}$ is ...
2
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0answers
43 views

How to prove the monotonicity of semi-norm?

In the book "Malliavin Calculus and related topics", the author states that $||F||_{k,p}=(E(|F|^p)+\sum_{n=1}^kE(||D^n F||^p_{H^k})^{\frac{1}{p}}$ has monotonicity property, i.e. $||F||_{k,p}\leq ||F||...
1
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1answer
65 views

How can I can calculate the CDF of the random variable?

Assuming that $H$, $N$ are random variables in which $H$ is distributed following exponential distribution with mean value $\Omega$, and $N$ is a Gaussian random variable with probability density ...
0
votes
1answer
198 views

Existence of a Continuous Modification of Fractional Brownian Motion

For a course on stochastic processes, I've been working on an exercise on fractional Brownian Motion. Showing that this process has a continuous modification is one of the final steps of the exercise, ...
2
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2answers
57 views

Independence of sums of infinitely many random variables

Let $\{X_i \}^\infty_1$ be independent random variables (that is, any finitely numbers of random variables are independent). Furthermore, if $S_n=\sum_1^n X_i$ converges almost surely to a random ...
0
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1answer
39 views

Determining 2 events in a sample space

Consider 2 events $A$ and $B$ in a sample space $S$. Assume that $\Pr(A) = {1\over 2}$ and $\Pr(B\mid\overline A)= {3\over 5}$. Determine $\Pr(A\cup B)$ Assume that $\Pr(A \cup B) = {5\over 6}$ and $...
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1answer
166 views

3 urns, each with 4 balls. select one ball from each

Three urns are labeled $1,2,3$. Each urn contains $4$ balls labeled $1,2,3,4$. A ball is drawn from each urn such that any ball is equally likely to be drawn. The number on the ball is compared to the ...
0
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1answer
78 views

Radon-Nikodym Derivatives between Ito Processes

I am curious about the following problem: Let $B_t$ be a standard Brownian motion on $(\Omega, \mathcal F, \mathcal F_t, \mathbb P_a)$, where the filtration is generated by $B_t$. On a finite ...