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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8 views

Finding the factorial moment generating function

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 ...
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0answers
7 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 ...
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0answers
24 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 ...
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1answer
9 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. ...
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0answers
16 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 ...
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2answers
26 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 ...
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1answer
21 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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1answer
46 views

Density of probability in a square [on hold]

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 ...
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2answers
22 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 ...
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1answer
18 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 ...
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2answers
22 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 ...
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1answer
28 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, ...
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2answers
29 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
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1answer
35 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 $X_0=0$. ...
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2answers
22 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 ...
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0answers
21 views

Probability density function of an uniformly distributed random variable [on hold]

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1).$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ for $\lambda>0$ And calculate $P(Y>t+s|Y>t)$ for ...
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2answers
106 views

Game of probability

n a game, played between $2$ players there is a circular field and one of the players is blindfolded, who stands in the center of the field. The other player stands at a fixed point on the ...
2
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1answer
22 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 ...
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2answers
31 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
40 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 ...
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1answer
30 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 ...
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1answer
16 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 ...
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1answer
31 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 ... ...
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2answers
17 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 ...
2
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1answer
28 views

Inequality for the derivative of a density of a random variable convolved with a normal r.v.

I have a question about the following proof. The statement is: Let $X$ be a random variable and $Z_\tau \sim N(0,\tau)$ be an independent random variable. Then $Y_\tau := X + Z_\tau$ has a ...
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0answers
30 views

How does Graham knows his number is really the upper bound to the dimension problem?

I know initially he stated that the answer is somewhere between 6 and Graham's number. How does he know that for Graham's number dimensions it is really impossible to color the lines that way? I know ...
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2answers
55 views

Borel set of $\mathbb R^n$ with $n > 1$

According to various sources, the Borel set over $\mathbb{R}^n$ can be defined in several equivalent ways: For instance, it can be defined as the smallest sigma-algebra containing every open set of ...
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1answer
45 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
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27 views

Probability dealing with Odds [on hold]

The odds that an event A will occur are given by the ratio P(A)/P(A'). Show that if the odds of an event A is a/b, then its probability is P(A) = a/(a+b).
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2answers
28 views

Probability of Winning a Toss

I have an unfair coin with two sides 1 and 2. I have a problem and its constraints. The coin has to be tossed until I win; which happens when 1 shows up in a toss. Constraints: Since the coin in ...
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1answer
18 views

Motivation behind the definition of the Itô integral for elementary predictable processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$. A real-valued stochastic process $H=(H_t)_{t\ge 0}$ is called elementary ...
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1answer
26 views

Proving technique used to show an equivalence to the definition of a Markov process

Let $X=(X_t)_{t\in I}$ be Markov process with values in a Polish space $E$. I want to show, that there exists a stochastic kernel $\kappa:E\times\mathcal{B}(E)^{\otimes I}\to [0,1]$ such that ...
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2answers
37 views

Tail probability of a random variable

Here are two theorems about the "tail probability" of a random variable X Thm1: The expectation $E(|X|^\alpha) < \infty$ for some positive $\alpha$ if and only if $$\sum_{n=1}^\infty ...
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2answers
25 views

Lebesgue Integral of an Indicator Function using Measure Theory

Let $X$ be a random variable on $\Omega$ and fix $c \in \mathbb{R}$. I recently saw the following in a calculation: $$ \int_{\Omega} \mathbb{I}_{(c,\infty)}(X(\omega)) dP(\omega) = P(X \geq c). $$ I ...
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1answer
34 views

(Elementary) Markov property of the Brownian motion

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 ...
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0answers
13 views

Definition of expected value of a continuous random variable [duplicate]

Let $X$ be a random variable with the probability desntiy function $f$. Then, according to the book "Intro to probability and statistics" by Rohatgi, the expected value of $X$ is defined as: $$E(X) ...
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23 views

A property of the space of rcll functions

Let $E$ be a locally compact separable metric space, $$\Omega=\{f:[0,\infty]\to E\mid f \text{ is rcll}\}$$ and $P$ be a probability on $\Omega$ which equiped the $\sigma$-algebra generated by ...
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1answer
40 views

Continuous mapping theorem - counterexample

The continuous mapping theorem states that Let $g: R^n \rightarrow R^k $ be continuous in every point of a set $C$ such that $\mathbb P\left(X\in C\right)=1$. If $X_n \xrightarrow{d} X $ then ...
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1answer
36 views

Computing conditional expectation $\mathbb E(U^V \mid U)$

Let $U$ and $V$ be iid uniformly continuous on $[0,1]$. How can I compute $\mathbb E(U^V\mid U)$? Which property do I have to use?
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1answer
27 views

Prove that the increments of the Brownian motion are normally distributed

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 ...
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0answers
52 views

Distribution of bounded summation of i.i.d random variables

We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ ...
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1answer
22 views

How to show that $\mathbb{E}(\lim_{n \to \infty} X_n) = 0$ when $X_n(x) := n \cdot 1_{[0,\frac{1}{n}]}(x) \qquad (x \in [0,1])$

from the answer of Exchanging limit and expectation for $L^2$ random variables: Consider for example the probability space $(\Omega,\mathcal{A},\mathbb{P}) := ...
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0answers
24 views

Closeness in distribution implies closeness in statistics?

I am aware that convergence in distribution does not necessarily imply convergence in the mean. I browsed through some examples of statistical distances here ...
2
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1answer
23 views

Almost sure convergence of $\hat{\sigma^2}$

Let $Y \sim N(X\beta,\sigma^2I)$ where $Rank(X_{n\times p})=p \leq n$. The least square estimate of $\sigma^2$ is $\hat{\sigma^2}=\frac{Y'(I-P)Y}{n-p}$ where $P=X(X'X)^{-1}X'$ is the projection matrix ...
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1answer
20 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that ...
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1answer
22 views

Distribution/law of a random variable after conditioning on an event

I have a probability triple $(\Omega,\mathcal{B},P)$ and a random variable $X:(\Omega,\mathcal{B},P)\to(\mathbb R,\mathcal{R})$ with distribution $\mu_X := P \circ X^{-1}$. If I condition on an event ...
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1answer
77 views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
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2answers
20 views

let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is?

I have a question about probability: let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is? can someone ...
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1answer
19 views

Computation of Conditional Expectation using Measures

Here's a definition of conditional expectation of $X$ found on p 363 of this book: Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X$ be in integrable random variable. If $B \in ...
4
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4answers
85 views

Is it possible to assign probability to a set $X$ with $|X|>2^{\aleph_0}$?

Is it possible to assign probability to a set $X$ with cardinality $|X| > 2^{\aleph_0}$? Example would be a set $|X| = 2^{2^{\aleph_0}}$.