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 ...

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51 views

From distribution function to probability measure

So assume we have a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X : \Omega \rightarrow \mathbb{R}^*$. We can derive from this a distribution $P_X$, and a distribution function ...
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1answer
127 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
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1answer
83 views

Order related to Empirical distribution function and Normal distribution

Let $X_1,\dots,X_n$ are i.i.d with distribution function $F$. Let $\hat F_n$ be its empirical distribution function, i.e., $$ \hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x) $$ where $1_A(x)$ is the ...
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1answer
57 views

Help with a problem involving limit theorems

Here's a problem I am stuck on. The problem goes as follows: Suppose the distribution of scores of a test has mean 100 and standard deviation 16. Calculate an upper bound for the probability ...
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160 views

Compare two estimators by using the their Expected value and variances

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
73 views

Prove that $COV(h(x),g(x)) \leq 0$ means the different direction for $h,g$

(Covariance Inequality) Prove that if $g$ is nondecreasing and $h$ nonincreasing, then $$ E(g(X)h(X)) \leq E(g(X)) E(h(X)) $$ I know that it is equivallent to prove $COV(g(X),h(X)) \leq 0$ if $h$ ...
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1answer
33 views

Given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, prove that $P(C\mid A)≥1-\varepsilon$

We need to show that, given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, that $P(C\mid A)≥1-\varepsilon$. Since we know that $P(C\mid B)=1$, it follows that $P(B ...
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1answer
72 views

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds:

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds: $P(Z\leq -u \space \text{or} \space Z\geq v) \leq \frac{4\sigma^{2} + ...
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63 views

What is the intuitive meaning of $K_1, K_2, K_3$ in regards to the conditional density formula derivation in Brownian motion.

In my text, there is a passage that says: "Suppose we require the conditional distribution of $X(s)$ given that $X(t) = B$, where $s < t$. The conditional density is: $$ \begin{align*} f_{s\mid ...
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444 views

complex integration over the whole plane

I am trying to solve this integral: $H(z)=\int_{\mathbb{C}}{p(z)\log_2{p(z)}dz}$, where $z$ is a complex number with complex normal distribution $p(z)$, and $\mathbb{C}$ denoted the complex plain. ...
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1answer
436 views

probability question on characteristic function

I got a big problem with my exam practice question on characteristic function. Here are two. Let $X$, $Y$ be two independent random variables with the following characteristic functions: ...
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1answer
139 views

Continuity of Expected Value

Let $m(\cdot)$ be a probability measure on $Z$, so that $\int_Z m(dz) = 1$. Consider a continuous function $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subseteq \mathbb{R}^n$, ...
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0answers
112 views

When can a measurable mapping be factorized?

Problem 13.3 of Probability and Measure by Billingsley states: $(\Omega, \mathcal{F})$ and $(\Omega', \mathcal{F}')$ are two measurable spaces. Suppose that $f: \Omega \rightarrow ...
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1answer
87 views

Covariance inequality for $n$ exchangeable random variables

Let $n \in \mathbb{N}$, $n \geq 2$, assume that $X_1,\ldots, X_n$ are exchangeable, square integrable random variables with $\mathbf{E}\bigl[X^2_1\bigr] < \infty$. Prove that the following ...
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1answer
96 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
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1answer
91 views

Unifying the treatment of discrete and continuous random variable

I have been working on the reconciliation of the treatment of discrete and continuous random variable in a measure theoretic sense. But I found myself blocked on fundamental results. We know that If ...
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1answer
51 views

Definition of the set of independent r.v. with second moment contstraint

I am trying to nice write the definition of the following set. Def: The set of all distributing of the pair $(X_1,X_2)$ such that $X_1$ and $X_2$ are independent Have second moment constraint ...
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2answers
70 views

Simple formula on $X_n^{(k)} = \sum_{1 \le i_1 < … < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$ (to show $X_n^{(k)}$ is martingale)

Let $$X_n^{(k)} = \sum_{1 \le i_1 < ... < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$$ If I take $k=2$ and $S_n = Y_1 + \dots + Y_n$ I have of course: $$X_n^{(2)} = \frac{1}{2} (S_n^2 - ...
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1answer
70 views

For $E (X - EX)^2$ to exist, do we need $EX$ to exist and be finite?

For $E (X - EX)^2$ to exist (may be infinite), according to $E (X - EX)^2 = E X^2 - (EX)^2$, I think a necessary and sufficient condition is $EX$ exists and is finite, because $ E X^2 \geq ...
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78 views

integral with respect to Dirac measure

Let $\delta_a(A)$ be the Dirac measure, that is $\delta_a(A) = 0$ if $a \notin A$ and $\delta_a(A) = 1$ if $a \in A$ and $\phi : \mathbb{R} \rightarrow \mathbb{R}$ a bounded Borel function. What does ...
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1answer
54 views

Mutual Independence Definition Clarificaiton

Let $Y_1, Y_2, ..., Y_n$ be iid random variables and $B_1, B_2, ..., B_n$ be Borel sets. It follows that $P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$...I think? If so, does ...
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1answer
39 views

Application of Kolmogorov's Zero-One Law

Most of the books I read, they only state some examples of tail events. One of them is $[\sum_n X_n \ converges]$. My main problem is to show that this is indeed a tail event.
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1answer
70 views

Brownian motion - Hölder continuity

Let $B$ stand for a Brownian motion on a finite interval $[0,1]$. If I am not wrong, I think that there exists a positive constant $c$, such that almost surely, for $h$ small enough , for all $0< t ...
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1answer
33 views

Choosing the correct subsequence of events s.t. sum of probabilities of events diverge

Here is the problem. I tried choosing $B_n = A_{mn}$ since it is an independent sequence for $m \geq 2$, but I am not quite sure how to guarantee that $\sum_{n=1}^{\infty} P(A_{mn}) = \infty$. Is ...
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1answer
142 views

$P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$

Let $\mathfrak{F} = (A_n)_{n \in \mathbb{N}}$. Prove that $P(\limsup A_n)=1$ if $\forall A \in \mathfrak{F}$ s.t. $P(A) > 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$. (Side question 1 Is second ...
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1answer
39 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
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2answers
74 views

Probability Theory - Fair dice

Three fair six-sided dice are thrown and the dice show three different numbers. Find the probability that at least one six is obtained. Im unsure ofwhat type of question this is, I have tried ...
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1answer
62 views

A basic question on expectation of distribution composed random variables

Suppose that $X$ and $Y$ are random variables with distribution functions $F$ and $G$. If $F$ and $G$ have no common jumps then I need to show that $E[F(Y)] + E[G(X)] = 1$. How to proceed here ? ...
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2answers
46 views

PDF of the addition of several outcomes from Poisson distribution

We draw $n$ values from a Poisson distribution and add them. - What is the expected of this addition - What is the PDF of this addition It seems quite intuitive to me that if we add $n$ Poisson ...
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1answer
80 views

Expected Value and Variance of Two Random Variable

Let $X_1, X_2,...,X_n$ and $Y_1,Y_2,...,Y_m$ be independent exponential distributed random samples with mean $\theta$. Let $T\alpha = \alpha\bar{x} + (1-\alpha)\bar{y}$, where $0 < \alpha < 1$. ...
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1answer
67 views

Approximate Expectation of $x^2$

I am estimating the $E[x^2]$. The function I am looking at gives you a constant value of A from 0 to 4, it is 0 from 4 to 6, and A from 6 to 10. I got that $E[x]$ to be 5. I calculated the value of ...
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1answer
89 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
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0answers
86 views

Probability of convergence of a monotone sequence

Let $\left(\Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space, and let $X: \Omega \rightarrow \mathbb{R}^n$ be a random variable. Let $\mathbb{P}^N$ denote the product measure $\mathbb{P} ...
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1answer
342 views

Expected value of c.d.f when normal distributed

I need help to calculate the expected value of an invertal of a c.d.f function which is normal distributed. I know that $E(X)=\int^\infty_0 (1-F(x))dx$ What i need is to calculate $E(w|w \geq ...
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1answer
1k views

What is the PDF of a product of a continuous random variable and a discrete random variable?

Let $N$ be a discrete random variable which takes values in [0, ..., M], M > 0, with known PDF $P(N=n)$. Let also the continuous random variable $Z = \sum_{i=1}^{N}X_{i}$ as the sum of i.i.d. $X_{i}$ ...
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0answers
139 views

How to model mutual independence in Bayesian Networks?

It's well known that 3 random variables may be pairwise statistically independent but not mutually independent, for an illustration see: example pairwise vs. mutual relations. Can mutual ...
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1answer
93 views

Expression for $n$-th moment

I stumbled upon an expression in an article of statistics for an $n$-th moment with $X$ being a random variable over $[0, \infty)$. $$\mathbb{E} X^{n} = \int^{\infty}_{0} nz^{n-1}\; \text{Pr}(X > ...
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1answer
247 views

Probability of two opposite events

Suppose there is string of eight bits, e.g.: 00100110 Bits are randomly chosen from the string. All choices are done equally likely. Probability of choosing $0$: $p_0 = \frac{5}{8} = 0.625$ ...
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2answers
695 views

Minkowski's Inequality For Infinity

I've tried figuring this out and searching the net on this for 5 hours, but I can't get it. Every source says it's trivial, but I must be missing something because I have pages of work that don't lead ...
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97 views

Covariance inequality for infinitely many exchangeable random variables

Let $X_1, X_2, X_3,\ldots$ be exchangeable, square-integrable random variables. Show that $\mathbf{Cov}[X_1, X_2] \geq 0$. Solution: Assume that \begin{equation*} \mathbf{Cov}[X_1, X_2] ...
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1answer
2k views

Finding an expression for the probability that one random variable is less than another, given a condition.

Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as: $$\mathbb{P}[X < Y] = ...
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2answers
748 views

On the set of the sub-sums of a given series

Choose a sequence $(x_n)_{n\in\mathbb N}$ of nonnegative real numbers with finite sum $x=\sum\limits_{n\in\mathbb N}x_n$ and consider the set $X=\{x_I\mid I\subseteq \mathbb N\}$ where, for every ...
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4answers
4k views

Where to begin in approaching Stochastic Calculus?

I have experience in Abstract algebra (up to Galois theory), Real Analysis(baby Rudin except for the measure integral) and probability theory up to Brownian motion(non-rigorous treatment). Is there a ...
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3answers
2k views

The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is ...
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4answers
3k views

Questions on atoms of a measure

In Kai Lai Chung's A course in probability theory, An atom of any probability measure $\mu$ on $(\mathbb{R}, \mathcal{B})$ is a singleton $\{x\}$ such that $\mu({x}) > 0$. In Wikipedia: ...
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1answer
2k views

What are some open research problems in Stochastic Processes?

I was wondering, what are some of the open problems in the domain of Stochastic Processes. By Stochastic Processes. Any examples or recent papers or similar would be appreciated. The motivation for ...
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2answers
767 views

Cover time chess board (king)

Consider a random walk of a king on a standard chess board, which at each step moves to a uniformly random permitted square. What's the exact mean time to visit all squares (cover time), starting ...
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2answers
966 views

Translations of Kolmogorov Student Olympiads in Probability Theory

I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward. I ...
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1answer
352 views

Definition of the Brownian motion

The way I understood the definition of a Brownian motion $B_t$ in $\mathbb R$ is that it consists of two parts: We first define the finite-dimensional distributions $$ ...
6
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1answer
207 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...