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 about Inclusion-exclusion formula?

The problem requires to use the indicate function to prove the Inclusion-exclusion formula. But I really don't know what to do. Anyone can help with that? Thanks!
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Clarification of Proof on Kac's Theorem for Characteristic Functions

There is a proof given here that I don't really understand, and was hoping someone more competent could explain it in some more detail: Moment generating functions/ Characteristic functions of $X,Y$ ...
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22 views

Analog of the Chebyshev's inequality

Lets consider two random variables $\xi$ and $\eta$, which satisfy such conditions: $$E\xi = E\eta = 0,~~~ D\xi^2 = D\eta^2 = 1;$$ $$\operatorname{cov}(\xi, \eta) = \rho.$$ How can we prove? ...
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1answer
21 views

How to prove this expectation equality?

How to prove this expectation equality? I am studying probability theory by myself and I find it hard. Thanks!
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1answer
18 views

Constraints on correlation coefficients of multiple random variables

I am looking for a generalization of Correlation between three variables question for more than three variables. It is stated in one of the answers there that, for three variables with pairwise ...
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17 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
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3answers
63 views

How to prove $E(\sum\limits_{i=1}^\infty X_n)=\sum\limits_{i=1}^\infty EX_n$

How to prove that if $X_n>0$, then $E(\sum\limits_{i=1}^\infty X_n)=\sum\limits_{i=1}^\infty EX_n$? I think I should use something like monotone convergence theorem, but I really don't know how to ...
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1answer
31 views

How to prove $EX_n \uparrow EX$? [on hold]

How to prove that if $EX^-<\infty$ and $X_n \uparrow X$ then $EX_n \uparrow EX$? Can someone help me with this problem? Thanks so much!
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30 views

Proofs about expectation equality? [on hold]

How to prove the equality? Can anyone help me? THANKS SO MUCH!
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1answer
35 views

How to prove the inequality using Jensen's inequlaity?

How to prove the above inequality? I am learning probability by myself and it has been confusing me for days. Thanks!
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3answers
27 views

How to prove the above expectation inequality?

If $\mathbb{E}[|X|^k]<\infty$ then for $0<j<k$, $\mathbb{E}[|X|^j]<\infty$, and furthermore $\mathbb{E}[|X|^j]\leq(\mathbb{E}[|X|^k])^{j/k}.$ How to prove the above expectation ...
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4 views

Measure to compare quality of synthetic data generated?

What is a good measure to compare the quality of the synthetic data generated with respect to the original data? The synthetic data I have, is the scaled up version of the original data. I am confused ...
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1answer
17 views

Multiplicativity of expectation for real functions implies the same for complex-valued

Suppose that $$\mathbb{E} [f_1(X_1) \ldots f_M(X_M)] = \mathbb{E}[f_1(X_1)]\ldots\mathbb{E}[f_M(X_M)]$$ for all functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Is it necessarily true that this also ...
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41 views

Counting paths from the origin to a given point

Consider the following "walk" from the origin (0,0) in the plane to the point E=(5,5). A walk consists of starting at the origin and on each move, moving either one unit distance up or one unit ...
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1answer
20 views

Computing the expectation of conditional variance in 2 ways

Same as here. Let $X$ be a square integrable random variable on $(\Omega,\mathcal{F},P)$. Let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Define the conditional variance of $X$ given ...
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24 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
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2answers
27 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
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19 views

Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
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1answer
33 views

E[X|Y]=E[X] When X and Y are Independent — Proof from Book Question

I'm trying to understand a proof in a book for the following Theorem: Let $Y \in L^1 (\Omega, A, P)$ and suppose $X$ and $Y$ are independent. Then $E[Y|X]=E[Y]$. Proof: Let $g$ be bounded Borel. ...
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1answer
18 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
3
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1answer
42 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
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1answer
33 views

Pdf of the product of an exponential rv and a $f_Y=Ka^{-K}y^{K-1}$ distributed rv …

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ has the following Pdf: $f_Y=Ka^{-K}y^{K-1}, 0 \le y \le a $. Someone claims ...
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21 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
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18 views

(Statistics)Probability of given sum in dice tossing [on hold]

I need some help with this problem: By tossing two dice, what is the probability of: i) Total sum of 7 ii) Difference of 5 iii) Total sum multiple of 7 Thanks everyone ~Chris
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46 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
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16 views

Deriving independence relation properties

I have a small question regarding the derivation of an independence property. Suppose I have the following: $$I_{Pr}\left(X, Z, Y \cup W \right) \leftrightarrow P\left(C_{X}\mid C_{Z} \wedge C_{Y ...
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15 views

Show that the Bernstein polynomials f approximate f uniformly. [on hold]

Bernstein basis polynomials ${b_{i,n}}, 0\leq i\leq n$ defined as ${b_{i,n}}:[0,1] \rightarrow \mathbb{R}, x \mapsto \binom{n}{i}x^i(1-x)^{n-1}$. For a continuous function $f: [0,1] \rightarrow ...
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1answer
79 views

Uniform distribution with random support

I have a uniform distribution $ X \sim U(A,B) $ where the limits themselves are random: $A \sim N(\mu_A,\sigma_A^2)$ and $B \sim N(\mu_B,\sigma_B^2)$. Hence the support of $X$ is random. $A$, $B$ are ...
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1answer
19 views

Showing something converges, in distribution, to a normal distribution

I'm not sure how relevant the first few parts are, but I will post it just in case... $(X_i,Y_i), i=1,\dots,n$ are independent where $X_i$ has an exponential distribution $\mathcal{E}(\lambda_i)$ ...
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1answer
12 views

Order between probability measures: sets full below

Consider a product space $X = \{0,1\}^\mathbb{Z}$ and the space of probability measures on $X$, $\mathcal{M}(X)$. We say that for any two $a, b \in X$, $$a \prec b \iff a_x \leq b_x \, \, \, \, \, ...
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2answers
43 views

Well-definedness of the characteristic function of a compound Poisson variable

I am reading about compound Poisson variables and cannot get through the following statement. Let $\nu$ be a non-zero finite measure on $\mathbb{R}\setminus\{0\}$. Assume that $$\int ...
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37 views

A simple question about probability that a r.v. is less than another

I kept on getting an answer that I don't want, and I believe it hinges on this (really) simple question that I seem unable to verify because I am too tired... We know that if $X$ and $Y$ are ...
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1answer
81 views

How to prove a stochastic process does not exist?

I'm studying the stochastic analysis material, and stuck with a problem which states below: Suppose $\{X_t, 0 \leq t \leq 1 \}$ is a real-valued stochastic process that satisfies (a) $X_s$ and ...
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1answer
20 views

Angle of three independently chosen points on a cricle

If a,b,c are three points on a circle (viewed in $\mathbb{R}^2$ not disc) chosen independently and uniformly,and p(x) is the probability that at least one of the angles of the triangle formed by the ...
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1answer
18 views

Expectation of a Random Variable multiplied by a Conditonal Expectation.

In a probability book I'm reading (Jacod and Protter) it states: Let $Y \in L^2(\Omega,A,P)$, and $\mathscr{G}$ a sub $\sigma$ algebra of $A$. Then the conditional expectation of $Y$ given ...
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17 views

Expected value and General distribution Function

Given an iid random variable, the expected value is usually defined as follow: $E[T]=\int_0^\infty t f(t) dt $ where $f(t)$ is the pdf of $T$. On a book I found this definition (without any other ...
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34 views

Show that $E[X|\sigma (\mathscr{G},\mathscr{H})] = E[X|\mathscr{G}]$ if $\mathscr{H}$ independent of $\sigma (\sigma (X),\mathscr{G})$

The problem I'm working on is: If $X \in L^1(\Omega,\mathscr{F},P)$, and $\mathscr{G},\mathscr{H}$ are $\sigma$ sub-algebras of $\mathscr{F}$, with $\mathscr{H}$ independent of $\sigma (\sigma ...
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1answer
7 views

Markov Processes: $P_x$ and $E_x$

In the study of Markov processes, one usually introduces the measures $P_{\pi}$ on the path space of the process where $\pi$ is an initial distribution of the process $X$ i.e $\pi=\mathcal L(X_0)$. ...
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1answer
15 views

Proof all possible unions of a collection of sets is a sigma algebra

If one lets $\Omega$ be a probability space, and S = $\{A_1, A_2, ... \}$ be a collection of subsets of $\Omega$, then I would like to prove that the set of all possible unions of elements in S is a ...
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7 views

Equivalent Formulation of Markov Property for Homogeneous Chains

In Shiryaev's Probability (just above the strong Markov property, p.568), the author says that an equivalent formulation of the usual Markov property for homogeneous chains is $$P[\theta_nX\in B\mid ...
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11 views

Strong Markov Property for Discrete Stopping Times

I'm having a hard time deciphering a particular proof of the following strong Markov property. Theorem (Strong Markov Property) Let $X$ be a time homogenous Markov process with $T=\mathbb ...
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1answer
30 views

Problem with topological space in probability theory. [on hold]

Let $(X, \tau)$ be a topological space. a) Show that arbitrary intersections of closed sets are closed. b) Prove that a set $F \subseteq X$ is closed if and only if for all sequences $\{x_{n}\} ...
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35 views

Combining large number of independent probabilities

I am trying to calculate likelihood of laser scan($Z$) at give pose($x$) with known map ($m$) using beam based model $P\left(z_t|x_t,m \right)=\prod_{i=1}^{n}P'\left(z_i|x_t,m \right)$ My scan ...
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1answer
17 views

$n$-step transition probability of a Markov chain

Let $(X_t)_{t\in\mathbb{N}_0}$ be a time-homogenous Markov chain over a finite state space $\left\{1,\dots,m\right\}$, so that we've got $$\Pr(X_{t+1}=j\mid X_t=i_t,\dots,X_0=i_0)=\Pr(X_{t+1}=j\mid ...
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29 views

Comparing Squared Difference of a Random Variable and it's mean, or it's mean conditioned on a $\sigma$-field.

I have $X \in L^2$, and I want to show $E[(X-E[X|G])^2] \leq E[(X-E[X])^2]$ All I've tried is expanding and simplifying, which gives me wanting to show: $E[E[X|G]^2]-2E[XE[X|G]] \leq E[E[X]^2] - ...
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1answer
44 views

What is the bound on $E\|Y_n\|^4$ in terms of $n$?

Let $X_n,n\in\mathbb{N}$ be i.i.d. zero-mean random variables in some separable Hilbert space with $E\|X_n\|^8<\infty$ and $Y_n=\frac{1}{n}\sum_{i=1}^nX_n$. I need to find bounds on $E\|Y_n\|^4$. ...
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1answer
17 views

The “on $\left\{ \tau <\infty \right\}$” in the Strong Markov Property

The strong Markov property is often formulated as $$P[\theta _{\tau}X\in A\mid \mathscr F_{\tau}]\overset{\text {a.s on }\left\{ \tau <\infty \right\} }{=}P_{X_\tau}(X\in A)$$ What exactly does ...
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13 views

Expectation in Hidden markov process [on hold]

Let the hidden state variable be $S_t\in \{0,1\}$, and the observed variable be $X_t$. How can we prove that $\mathbb{E}(P(S_t=1|\sigma(X_{t-1},...,X_0)))= \frac{p_{01}}{p_{01}+p_{10}}$, where ...
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19 views

Conditional probability in multinomial distribution

Consider a multinomial distribution with $r$ different outcomes, where the $i$th outcome having the probability $p_i$, $i$=1,...,$r$, $\sum_{i=1}^r p_i = 1$. Denote $X_i$ be the number of times the ...
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+100

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...