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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31 views

Is random vector a vector of observations or list of random variables.

I am stuck on a very basic concept in probabilities. The question goes like this. Is a vector of observations of a random variable (for example a series of coin toss states) the same as a random ...
1
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0answers
25 views

Mutual information as a fraction of entropy?

Suppose I have two (discrete) random variables $(X,Y)$ with some joint distribution $P$. The mutual information $I(X;Y)$ is informally defined as the reduction is the remaining entropy in $X$ once the ...
7
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1answer
106 views

need reference for fact about products of measures

Need a reference (textbook or paper) for the following (probably well known) fact: Suppose $(X,M)$ is a measurable space and $\lambda$, $\nu$ are two different probability measures on $(X,M)$. Write ...
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3answers
28 views

Independence of intersections

Let $(\Omega,\mathfrak{A},P)$ be a probability space and $A,B,C\in\mathfrak{A}$ some events where $A$ and $B$ are independent. I'm a bit confused now as I intuitively think that this also implies ...
3
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1answer
36 views

Stochastic process $\exp(W_t - t/2)$ approaches zero for large $t$, but it is a martingale?

The stochastic process $$ S_t = \exp\left( W_t - \frac{1}{2} t \right) $$ is a martingale (for example this could be seen by noting that it solves the SDE $dS_t = S_t dB_t$, which has no drift). But ...
3
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1answer
42 views

Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on ...
1
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1answer
44 views

example computing expectation

I am trying to understand the following example: A fair die is rolled, and whichever number comes up, a fair coin is then flipped that many times. Let $N$ be the outcome of the die roll, and $X$ the ...
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0answers
14 views

Convergence in “the” Skorohod topology for monotone functions

Let $x_n$, $x$ be nondecreasing cadlag functions on $[0,T]$. To get $x_n\to x$ in Skorokhods $M_1$-topology, we only have to prove convergence on a dense subset of $[0,T]$ including 0. (see Whitt: ...
2
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2answers
43 views

Expectation of the maximum as the number of random variables goes to infinity?

Suppose $v_1,v_2,...,v_n$ are $n$ i.i.d. continuous random variables with the range $[\underline v,\bar v]$, does the expectation of the maximum of the random variables $E[\max v_i, i=1,2,\dots,n]$ go ...
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2answers
42 views

Conditional Probability $P(B| A\cap B) = 1$?

$P(B|A\cap B) = 1$? I'm a little confused about the probability of $B$ occurring given that $B\cap A$ occurred. If $B\cap A$ happens, does this guarantee $B$'s occurrence or must we consider the ...
1
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1answer
54 views

Itô integral with respect to a diffusion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
1
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1answer
60 views

How does one define the Fourier transform of a probability distribution?

Say $p_X$ and $p_Y$ are two probability distributions on a $m$ element set. Then I see an equality written as, $$\sqrt{m} \vert \vert p_X - p_Y \vert \vert _2 = \sqrt{ \sum_{k=0}^{m-1} \vert ...
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1answer
25 views

Equivalent definitions for expected value of random variable

Let $(\Omega, P)$ be a probability space. One definition for the expected value of a random variable $X$ is $$E(X)=\sum_{x\in \mathbb{R}} xP(X=x).$$ The notes I am reading say that this definition ...
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1answer
62 views

Proof that random variable is almost surely constant

If a random variable $X : \Omega \to \mathbb R$ is $\{ \emptyset, \Omega \}$-measurable, then it is constant. I want to generalise this result: Now if $\mathcal G$ is a $\sigma$-algebra such that ...
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3answers
45 views

Book to learn Mathematical Probability theory? [closed]

What are some good references to , good book to learn Mathematical Probability theory ? Please help .
4
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1answer
28 views

Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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2answers
40 views

Calculating a characteristic function two different ways gives contradictory results. Why?

I am trying to calculate a characteristic function directly and via the conditional distributions. I get contradictory results: Let $X$ and $Y$ be random variables defined on the same probability ...
0
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1answer
28 views

Matrix Multiplication, Trace and Integration

Let $\omega(x)$ be a $p\times 1$ vector-valued function defined on a random variable $X$ with CDF $F$. Now define $$V:=\int \omega(x)[\omega(x)]^T dF(x).$$ Then define $\gamma$ as follows. $$ \gamma ...
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1answer
23 views

Prove that the Itô integral for elementary predictable processes builds a martingale

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
0
votes
1answer
38 views

why does $P(X_{n+m} = j \mid X_m = k, X_0 = i) = P(X_{n+m}= j \mid X_m = k)$ follows from the Markov Property

I've learned that the Markov property says the following: $$P(X_{n+1} = i \mid X_n = j, X_{n-1}= j_1,\dots, X_0 = j_n) = P(X_{n+1}= i \mid X_n = j)$$ For me it is not clear how you can derive the ...
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1answer
13 views

Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
1
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1answer
26 views

Comparing Percentiles of 2 Samples Drawn from the Same Distribution

Suppose I have two sets of numbers: $A=\{a_1,a_2,...a_{N_1}\}$ and $B=\{b_1,b_2,...b_{N_2}\}$ with $N_1<N_2$. WLOG assume that $a_i<a_j$ for all $i<j$ and similarly for $b_i$ and $b_j$. ...
2
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2answers
41 views

Probability of drawing white ball after transferring to new urn n times

I am in a probability theory course and could not find the solution to this question anywhere. The assignment is already turned in, and I am asking this for my knowledge and for others who are also ...
4
votes
1answer
59 views

Probability theory required for learning statistics rigorously

I would like to learn statistics rigorously. The only book that I can find that seems to do statistics rigorously is this book "Theory of statistics" by Schervish (which seems advanced): ...
1
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1answer
90 views

Probability of a rolling a dice $n$ times with $k$ faces

I need help calculating the probability of rolling $n$ dice with $k$ faces. So you have multiple dice, all with $k$ faces (number of sides on a dice) and you want to calculate the probability of a ...
1
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3answers
37 views

About discrete probabilities (Expected values)

Is my solution correct? Suppose two player (A and B) each one with 200,00 dollars toss a coin not balanced in a such way that the probability of head is $p$. Suppose yet that if the result obtained ...
2
votes
2answers
44 views

Does the distribution of a process on $\mathbb{R}^{[0,\infty)}$ uniquely define it?

Question: Can I have two different stochastic processes $(A_t)_{t \in [0, \infty)}$, $(B_t)_{t \in [0, \infty)}$ having the same distribution on $\mathbb{R}^{[0, \infty)}$ differ in some ways? ...
4
votes
0answers
41 views

Why is a predictable stochastic process called *predictable*?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I$ be an index set $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be a stochastic ...
0
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0answers
13 views

Finding the factorial moment generating function [duplicate]

I need help finding $G_x(t)$ $f(x)= pq^{x-1}$ for x = 1, 2,... and 0 otherwise. I know $G_x(t)= M_x(ln t)$ I have started the following $$\sum_{x=1}^\infty e^{xlnt}f(x)$$ $$\sum_{x=1}^\infty ...
0
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0answers
21 views

Probabilistic Modelling of uncertain positions of objects in a 2D-Grid

I have a 2D-Grid which is populated by obstacles of different sizes. A size is always a whole number of cells. An obstacle is at least one cell big. If I did kown the size of the object but had only ...
0
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0answers
38 views

What is really a probability distribution?

I have some difficulties understanding what is difference between probability measure and probability distribution? I have always thought that term probability distribution is reserved for measure on ...
0
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1answer
14 views

Optimize distributions for low mean, high variance

Assume a context with $N$ approximately normal distributions where a lower mean implies a 'better' distribution and a high variance or high standard deviation implies a 'better' distribution as well. ...
0
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0answers
26 views

distance distribution in Poisson point process

Consider a homogeneous Poisson point process in 2D space with density $\lambda$ per unit area. Let $\mathcal{B}(o,R)$ denote a disk centered at origin with radius $R$. Let $n$ be the number of points ...
1
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2answers
29 views

Expectation of Nonnegative Random Variable - Measurability

Recall the result that for a nonnegative random variable $X$ on $(\Omega, \mathcal{F}, P)$, $$ E[X] = \int_0^\infty (1 - F(x)) dx, $$ where $F$ is the cdf of $X$. In many of the proofs I've seen for ...
2
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1answer
30 views

Example of Non-separable stochastic process.

This question is related to the link: http://www.encyclopediaofmath.org/index.php/Separable_process The link provided a basic definition of separable Stochastic process. I felt all the process under ...
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votes
1answer
49 views

Density of probability in a square [closed]

Suppose we have a square $$\{(x,y) : x \in [0,1], y \in [0,1] \}.$$ We suppose that we have $X$ and $Y$ are the coordinates in this square that are uniformly distributed. Why does the joint density is ...
1
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2answers
24 views

Essential supremum via cumulant

Let $p(t)=\log \mathbb{E}[\exp (tX)]$ for $X$ real valued random variable. Now it holds (assuming that $p$ is smooth and finite on $\mathbb{R}$) that $p'(\infty)=\text{ess}\sup X$. How can I prove ...
1
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1answer
21 views

If $E(|X|)<\infty$, how do we show that it can be expressed as below

$F(x)$ is the distribution function of $\mathbb X$, and $f(x)$ is the derivation of $F(x)$, Prove that $\int_{0}^{\infty}(1-F(x))dx-\int_{-\infty}^{0}F(x)=E(X)$. Note that ...
1
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3answers
34 views

For non-negative data the sample mean is not smaller than its standard error.

(1) Let $X_1, X_2, \dots, X_n$ be a random sample from a population with non-negative values. Then show that $\bar X \ge S/\sqrt{n},$ where $S^2 = [\sum_{i=1}^n (X_i - \bar X)^2]/(n-1).$ I have not ...
2
votes
1answer
31 views

Why a change of a brownian motion does not depend on the past values of it? [duplicate]

$(B_t)_{t \in \mathbb R_0^+}$ are random variables on $(\Omega,\mathcal A,P)$. $\forall r \le s, t > s, B_t-B_s,B_r$ are independent (i.e. $\sigma(B_t-B_s)$ and $\sigma(B_r)$ are independent, ...
1
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2answers
36 views

Prove that $X,Y$ are independent iff the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $X$ and $Y$ be random variables on $(\Omega,\mathcal A,\operatorname P)$ with values in $\mathbb{R}^m$ and $\mathbb{R}^n$, ...
2
votes
1answer
42 views

Is this martingale identically zero?

Let $X_t$ be such that $X_t$ is bounded continuous martingale adapted to the filtration $\mathcal{F}_t$ such that $$\Bbb{E} \bigg[\int_0^t e^{X_s} \, d\langle X\rangle_s\bigg] = 0$$ and $X_0=0$. ...
1
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2answers
28 views

If $B$ is a BM and $\mathcal F_t=\sigma(B_s,s\le t)$, then $(B_{s+t}-B_t)_{s\ge 0}$ is independent of $\mathcal F_t^+:=\bigcap_{s>t}\mathcal F_s$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
2
votes
1answer
29 views

Show that $|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$

Consider the random variables $X$ and $Y$ defined in the same space $\Omega$. Show that $$|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$$ This question comes from an old test, I know that ...
0
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2answers
32 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
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1answer
42 views

Expectation of a random variable in terms of its distribution function

Here is a theorem on expectation of a random variable in terms of its distribution function Theroem: Let $X$ be a (continuous or discrete) non-negative random variable with distribution function ...
0
votes
1answer
31 views

Proof the statements

Proof the statements below i)If $P(A)=0$ and $B$ is any event, then $A$ and $B$ are independents ii)If $P(A)=1$ and $B$ is any event, then $A$ and $B$ are independents iii)The events ...
1
vote
1answer
40 views

Are random variables in a tail σ-algebra in the same probability space?

Let $X_1, X_2, ...$ be random variables. Define $\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, ...)$ and $\mathscr{T} = \bigcap_{n} \mathscr{T}_n$, the tail σ-algebra of $X_1, X_2, ...$. When defining a ...
2
votes
1answer
61 views

Are the running products of iid RVs independent?

Are the running products of iid RVs independent? Let $Y_0, Y_1, ...$ be independent random variables with $P(Y_n = 1) = P(Y_n = -1) = 1/2 \ \forall n = 0, 1, 2, ...$ (*) Define $X_n = Y_0 Y_1 ... ...
0
votes
2answers
20 views

Understanding different definitions of bayes theorem

I am taking course on probability and reading about bayes theorem. In Sheldon Ross' book, it given as $$P(E) = P(E|F)P(F) + P(E|F^C)P(F^C)$$ with note: Equation above states that the probability of ...