Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Proof of the progressiveness of a stopped progressive process

I have trouble understanding the proof of Proposition 2.18 in Karatzas/Shreve: Brownian Motion and Stochastic Calculus, which states that a $\mathcal{F_t}$-progressively measurable process $X_t$, ...
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10 views

Generalize result on independent RV to conditional independence

Here Independence and conditional expectation is stated that $E(f(X)g(Y))=E(f(X))E(g(Y))$ iff $E(h(X)|Y) = E(h(X))$. Now I'm wondering if this generalizes to independence in the conditional sense, ...
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Completeness of the space of random variables with bounded conditional first moment with respect t0 $\left\Vert \cdot\right\Vert _{2} $

Consider a probability space $\left(\Omega,\mathcal{F},P\right) $, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F} $. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right) $ be the space ...
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17 views

Convergence in distribution of the Sum Y/\sqrt{\lambda}

The question follows: ${[X_n]}_{n\geq1}$ is a sequence of independent rv's such that: $P(X_n=-1)=1/2$ $P(X_n=1)=1/2$ Let $N \in Po(\lambda)$ where N is independent of ${[X_n]}_{n\geq1}$. ...
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Determining a Mass Function to $P(X>k+1|X>k) = (k+1)/(k+2)$

I have been struggling with this exercise for a while now and I could a push into the right direction. The exercise is the following: Let $X$ be a random variable which may assume only positive ...
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1answer
26 views

Proof about independent random variables

Let $X_1,X_2,...$ be independent random variables with $P(X_n=1)=p_n$ and $P(X_n=0)=1-p_n$ Show that $X_n\rightarrow 0$ in probability if and only if $p_n\rightarrow0$, $X_n\rightarrow 0$ almost ...
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Uniform distribution, as a sum of biased Bernoulli trials.

Suppose that the probability of $x=0$ is $p$, and the probability of $x=1$ is $1-p=q$. Consider the random sequence $X=\{X_i\}_{i=1}^{\infty}$. We map this sequence by $C$ to a point in the interval ...
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Processes adapted to time changes

I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a ...
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$E[e_te_s\Delta B_t\Delta B_s]$ for $\Delta B_t$ Brownian motion increments and $e_t(\omega)$ a measurable function.

Let $\Delta B_j=B_{t_{j+1}}-B_{t_j}$ where $B_t$ is Brownian motion, and $e_i(\omega)$ measurable with respect to $\sigma(B_{t_i})$. In Oksendal's 'Stochastic Differential Equations' he states: $$ ...
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16 views

Bounding $\mathbb{E}(X_{i_1}\cdot \ldots \cdot X_{i_k}) $

Consider random variables $X_1,\ldots X_n$ with zero mean, variance at most $1$, $k$-wise independent $k\leq n $ and bounded: $|X_i|\leq C$ for some $C\geq 1$. If I assume $k$ is even, how can I ...
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Inequality on Entropy

Suppose $1\geq p_1 \geq \cdots \geq p_n > 0$ and $\sum_{i=1}^n{p_i} = 1$, I find $$2^H \geq \sum_{i=1}^n{i \cdot p_i}$$ by experiments where $H = -\sum_{i=1}^n{p_i \log_{2}{p_i}}$ is the entropy of ...
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1answer
25 views

uniform convergence of characteristic functions

Assume that a sequence of probability measures $\mu_n$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Prove that $\phi_n$ ...
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2answers
16 views

Is multiplication normally/binomially distributed?

I was thinking about the binomial formula in the context of coin flips and got to thinking about the reason that even though HHHHHHHHHH is just as likely to occur as a sequence as HHHHHTTTTT, 5 heads ...
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1answer
11 views

verifying whether a conditional density function is valid

I want to verify whether a given conditional probability function is valid or not. $\mathsf P(y\mid x)=\begin{cases}c\, e^{(-y/x)} & : y\geqslant 0, x>0;\\ 0 & ...
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1answer
18 views

Introduction to Measure-theoretic Probability George Roussas. example 4 page 1 [on hold]

I am reading Introduction to measure-theoretic Probability George Roussas. example 4 page 1 says: Let $\Omega$ be infinite (countably or not) and let $\mathcal{C}= \lbrace A \subseteq \Omega;A$ is ...
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1answer
19 views

Expected Return, Expected Value, and an Ito Process

I am reading John Hull's "Options, Futures, and Other Derivatives". I am currently in Ch. 31 on the HJM Model. Hull makes a statement which a need an explanation for. First, some notation. Let ...
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25 views

The expected value of a random vector when the X_is are independent

$ \DeclareMathOperator{\var}{var} \DeclareMathOperator{\cov}{cov} $ The components of a random vector $\mathbf{X} = [X_1, X_2, \ldots, X_N]^{\intercal}$ all have the same mean $E_X[X]$ and the same ...
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1answer
32 views

How can I prove that without further assumptions Chebyshev's Inequality can not be improved?

I have found some examples on the web for specific random variables such $X$ (a discrete type) with probabilities $1/8$, $3/4$ and $1/8$ at the points $x=-1,0,1$ with $\mu=0$ and $\sigma=1/2$. Then, ...
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A minimization question about the convexity of KL-divergence

Let $f_1$ be a continuous density function which is given and consider the closed ball around $f_1$: $$\mathcal{G}=\left\{g:\int g(x) \ln\frac{g(x)}{f_1(x)}\mathrm{d}x \leq \epsilon\right\}$$ ...
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1answer
8 views

Bernoulli trials case in probability

A fair die is tossed twice. About how many times would you expect to roll 3 or greater? So based on sequence of Bernoulli trials: P(exactly k successes in n trials) = C(n,k) p^k q^(n-k) where p = ...
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probability distribution function of two independent variables

Let $X$ be a random variable whose distribution function is $F_X(t)=3^{-t}$. Suppose that $Y$ is another random variable whose distribution function is $F_Y(t)=4^{-t}$. What is the probability that at ...
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Probality Theory [on hold]

I would like to ask 3 questions about Probality theory. 1) Is there any research being done in Probality theory? 2) Which are the best universities to master in Probality Theory? 3) What a carreer ...
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1answer
29 views

Expected Service Times for truncated exponential

I'm trying to solve a problem where all arriving items (arrival exponential $\lambda = 1/5$) are divided into into groups, those who are served within 5 units of time and those who have their service ...
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19 views

Soft: Interesting examples of applications of the Itô–Nisio theorem

I'm writing up a presentation on the Itô–Nisio theorem and I'm looking for a simple (nontrivial) example showcasing it. I was thinking of a 2-dimensional simple random walk, but that seems ...
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1answer
25 views

How to prove that the sequence of random variables converges to a random variable?

If $Z_1,Z_2,\cdots,Z_n$ are random variables such that $Z(\omega)=\lim_{n \to \infty}Z_n(\omega)$ exists $\forall \omega \in \Omega$, then $Z$ is also a random variable. I was reading a book on ...
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1answer
18 views

Conditional expectation constant on part of partition

I have a question about conditional expectation, while looking for the answer here on stackexchange I noticed that there are a few different definitions used, so I will first give the definitions I ...
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1answer
27 views

characteristic function differentiation

Let $\mu$ be a probability measure on $\mathbb{R}$. Then the characteristic function is: $$ \varphi: \mathbb{R} \rightarrow \mathbb{C} \;\;\ \varphi(t):=i\int_\mathbb{R} e^{itx}d\mu(x) $$ Prove with ...
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42 views

The probability that $3$ random points on the circumference form a right-angled triangle?

In my probability theory course, I dealt with a similar problem which asks for the probability that $3$ random points on the circumference of a circle lie on the same semi-circle. But it makes me ...
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37 views

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function. It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow ...
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Proving that Poisson distribution is well-defined

I'm trying to prove that a Poisson distribution is a well-defined probability distribution -- i.e. that the sum of probabilities over all possible values is one. Since the distribution takes on ...
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1answer
21 views

is it true that conditional expectation Y to X is a function of X?

I mean, is it true that $E(Y|X) = \phi(X)?$ if so, how should we derive the form of X?
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Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
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1answer
26 views

Expectation over 2 random variables, help needed

Hi I am new here and I hope I can get some help. My question is about taking expectation over random variables. Lets say I have two random variables $\Xi$ and $\theta$ where $\Xi$ is for example a ...
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1answer
54 views

In frequentism, does every event have a probability?

For an infinitely repeatable trial with event space $\Omega$, and an event $A\subseteq \Omega$, the frequentist probability of $A$ is defined: $P(A):= \lim_{n\rightarrow\infty} \frac{n_a}{n}$, where ...
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1answer
12 views

Probability of non repeated value in a set of vectors (with integer values) for any number in the same vector position.

Suppose a set with $m$ vectors ($m$ finite) defined by $V_{i} = (x_{vi1},x_{vi2},\dots,x_{vin})$, with $i \in \left\{1, 2, \dots, m \right\}$ and $2 \leq n \leq p$, for a given $p \in \mathbb{Z}$ ...
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32 views

Optimal Number of White Balls

There are C containers, B black balls and infinite number of white balls. Each container should have at least one ball. Each of the containers may contain any number of black and white balls. Action ...
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1answer
27 views

Paley Wiener stochastic integral

Sorry for the stupid question, no answers necessary anymore! let $(B_t)_{t\in [0,1]}$ be a standard Brownian motion and $F\in C[0,1]$ differentiable. Then the sequence (which is an easy version of ...
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24 views

Bounding the difference of random variables by coupling.

Suppose we have two probability densities differing by atmost $\delta$. Is it possible to use coupling to have two random variables with the above two densities differing by less than $\delta$? I ...
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35 views

$\tau$ is stopping time. Check if $\tau + 1$, $\tau - 1$, $\tau^2$ also are stopping time.

Suppose that $\tau$ is stopping time. Is is true that a) $\tau + 1$ b) $ \tau - 1 $ c) $ \tau^2 $ also are stopping time? My prove: a) Yes, because forall t we have $$\{ t: \quad \tau+1 \le t \} ...
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Inner dependence of Independent random vectors

If $X = (X_1,X_2)$ and $Y = (Y_1,Y_2)$ , $X$ and $Y$ are stochastically independent can $X_1$ and $Y_1$ be dependent?
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Find $E[Z_1 | aZ_1 + bZ_2]$

Let's $Z_1,Z_2$ be a random variable such that $EZ_1^2 < \infty$ and $EZ_2^2 < \infty$. Find $E[Z_1 | aZ_1 + bZ_2]$ where $a,b \in \mathbb{R}$. We don't know what is distribution of $Z_1$ and ...
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expectation approximation error

Let $X$ be a random variable with no mass taking values in $\mathbb{R}$, and $f:\mathbb{R}\mapsto\mathbb{R}$ be a "smooth" function. I want to approximate $\mathbb{E}[f(X)]$ with $\mathbb{E}[g(X)]$ ...
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1answer
20 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
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30 views

how to related a weakly convergent random variable with its k-th moment

Let $\{X_n\}$ be independent random sequence with zero mean and unit variance. Suppose $$S_n:=\sum_{m=1}^n \frac{X_m}{\sqrt{n}} \Rightarrow X\sim \mathcal{N}(0,1)$$ holds. (Here "$\Rightarrow$" ...
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1answer
22 views

Simple bounding question for an expectation with truncating function

Let $\{X_m\}$ be independent random sequence. I want to show the following result Given $E[X_m^2]:=\sigma^2 < \infty$ and $$0 = \mathop {\sup }\limits_m P\left( {\left| {{X_m}} \right| > ...
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36 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
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60 views

Requesting deeper understanding of binomial coefficient

I noticed that $\binom {52} 4$ * $\binom {48} 1$ is $5$ times that of $\binom {52} 5$. So for example, if we were to draw $4$ cards from a standard deck then draw $1$ more, we cannot just say there ...
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16 views

Probability distribtuions [on hold]

A 10 metre by 10 metre plot of land is divided into 100 equally sized squares. Suppose that 300 seeds are randomly scattered on the plot of land. Use a suitable approximation to find the probability ...
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1answer
38 views

$X$ normally distributed in $\mathbb R^n$ iff components $x_i$ normally distributed?

We've had the normal distribution today in class and I was thinking about the following: Let $X$ be normally distributed, $X\sim N(a,\Sigma)$ with a symmetric positive definite matrix $\Sigma$ and ...
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21 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...