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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15 views

Probability Game

In a game you pick 5 numbers 1-15 and 4 of those numbers (from 1-15) are winning numbers. What is the sample space? What is the probability that you get no winning number and finally what is the ...
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2answers
191 views

A problem about the limit of measure

Problem: If $\mu$ is a $\sigma$-finite measure on $(R,\mathscr{B}(R))$, then define $\mathscr{A}$ to be the collection of all $A\in\mathscr{B}(R)$ such that the following limit exists and is ...
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1answer
9 views

Does $E[g(Y)|X]$ belong to $L^2(X)?$ ($g$: scalar function)

Consider two random vectors $X$ and $Y$ of respective size $p$ and $q$ with values in $\mathcal{X}$ and $\mathcal{Y}$. Let $g:\mathbb{R}^q\to\mathbb{R}$ be a scalar function. The result I'm looking at ...
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1answer
30 views

Moment assumption for product of random variables

Let us say that $U$ is a random variable with finite second moments $E[U^2] < \infty$. Is there a moment condition on $T$, say $E|T|^k < \infty$ for some $k$, that guarantees $TU$ having finite ...
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2answers
23 views

Prove that a sequence of maps is a sequence of i.i.d. r.v.

I need an help with the following exercise. I have a probability space $([0,1], \mathcal B, dx)$, where we denote with $\mathcal B$ the borelian sets in $[0,1]$ and $dx$ is the Lebesgue measure. We ...
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1answer
19 views

Question about definition of independent discrete random variables.

In my lecture notes, I am given the definitions for: -the independence of two discrete random variables -the independence of a set of discrete random variables -the pairwise independence of a set ...
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0answers
61 views

Maximum of *Absolute Value* of a Random Walk

Suppose that $S_{n}$ is a simple random walk started from $S_{0}=0$. Denote $M_{n}^{*}$ to be the maximum absolute value of the walk in the first $n$ steps, i.e., $M_{n}^{*}=\max_{k\leq ...
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1answer
67 views

construct a counterexample in measure theory

Problem: Construct a $\sigma$-algebra $\mathscr{F}$ of subsets of $R$ such that no open interval is measurable with respect to $\mathscr{F}$, although any singleton $\{x\}$ is ($x\in R$). I tried to ...
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2answers
47 views

Approximation of the binomial distribution

Let $S_n=\dfrac{B_n - np}{\sqrt{n\cdot p\cdot (1-p)}}$ be a random variable which has the standardized binomial distribution. From Chebyshev's inequality I know that $$P(|S_n| \ge x) \le ...
2
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0answers
33 views

showing conditional independence

This is from Kallenberg. Suppose $(\xi, \eta) \stackrel{d}{=} (\xi, \zeta)$, where $\eta$ is $\zeta$-measurable. Then $\xi$ is conditionally independent of $\zeta$ given $\eta$. The hint is to ...
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1answer
49 views

Exponential distribution - lack of memory

I'm having some problems understanding the concept of lack of memory. Given the following problem: There are two people at a counter, the time it takes to serve a customer is exponentially ...
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1answer
20 views

Does the outcome of rolling a six-sided die satisfies the definition of a probability space?

I recently learned about the formal definition of a probability space $(\Omega,\mathcal{F},P)$. But I feel like my grasp of it is a little shaky. I would like to understand this definition in the ...
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1answer
37 views

$E|X-m|$ is minimised at $m$=median

For a continuous random variable $X$, I want to show that $E|X-m|$ is minimum implies $m$ is the median of the distribution. Assume that the distribution function is $F$ and the density function is ...
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0answers
44 views

Weak convergence of order statistics

I've encountered the following problem: Let $U_1,...,U_n$ be iid uniformly distributed on $[0,1]$ and let $U_{n(k_n)}$ denote the $k_n$-th order statistic where $k_n$ is chosen, s.t. ...
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1answer
48 views

Is $X_t$ a martingale?

Is $X_t=\int_0^t$ Wsds a martingale? I checked the conditional expectation and I have $$E[X_t/F_u]=E[\int_0^u Wsds/F_u]+E[\int_u^t Wsds/F_u]=\int_0^u Wsds+\int_u^t E[Ws]ds=X_u$$, so it does look like ...
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2answers
62 views

Limits of integration of manipulated probability distribution function

I have to compute $\mathbb E\left[e^X \right]$, where $X$ follows a uniform distribution on $[0,1]$. I have started by computing the probability density function of $Y = e^X$, by doing the following: ...
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0answers
73 views

Convergence of probability measures on the space of real Radon measures

Consider the space $C_c := C_c(\mathbb{R})$ of compactly supported continuous functions with the inductive limit topology and $C_c'$ its topological dual which can be identified with the space of real ...
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2answers
141 views

Convergence in distribution to derive the expectation convergence

If $X_n\longrightarrow X$ in distribution, $\mathbb{E}(X)\lt\infty$, Do we have the following conclusion: $\mathbb{E}(X_n)\longrightarrow\mathbb{E}(X)$?
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1answer
46 views

lim sup (liminf) (An\Ak)

I need to show that $\limsup_n \liminf_k A_n \cap A_k^c=\phi$. Thus $\bigcap_n\bigcup_{r\geq n} \bigcup_k \bigcap_{m\geq k} A_r\cap A_m^c=\phi$? I am trying to show that $\lim_n P(\liminf_k A_n ...
2
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1answer
51 views

Uniqueness in Doob's Decomposition Theorem

I'm a little uneasy about one step in the uniqueness proof for Doob's Decomposition Theorem. Let $(X_n)_{n \geq 0}$ be a submartingale, $(M_n)_{n \geq 0}$ a martingale, and $(A_n)_{n \geq 0}$ be an ...
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1answer
16 views

Uniquness theorem from Monotone class theorem

I need to show using monotone class theorem (MCT) that; If $\mathcal{F}$ is a field, $P_1,P_2$ are two probability measures on $\sigma(\mathcal{F})$, then if $P_1=P_2$ on $\mathcal{F}$ then $P_1=P_2$ ...
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1answer
22 views

show conditional expectations are equal in distribution

This is from Kallenberg. Let $(\xi, \eta) \stackrel{d}{=} (\tilde{\xi}, \tilde{\eta})$, with $\xi \in L^1$. Then $E[ \xi | \eta ] \stackrel{d}{=} E[ \tilde{\xi} | \tilde{\eta} ]$. The hint is to ...
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1answer
11 views

Modification of a local martingale

I am quite curious to know if the following is true, which comes up to my mind when reading a paper on SLE: For any local martingale $(X_t)_{t \geq 0}$ and stopping time $\tau$, is it true that $$ ...
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1answer
36 views

question about markov chain with the states

If I understand correctly, this can be used as the definition of transient/recurrent state. but I need help to start please guide me to solve
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0answers
37 views

probability distribution estimation from correlated samples

I am looking to solve the following estimation problem. Consider a blackbox where (given below) given an input X, its N observations are recorded as output. These observations are denoted by $Y_1, ...
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0answers
26 views

Proof for smallest sigma field to contain a class of subsets of the sample space of an experiment.

I've started studying theoretical probability and came across this statement that Given a class of subsets e of the sample space X of an experiment E, the intersection of all sigma fields of subsets ...
2
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1answer
46 views

Strict stationarity of a MA(1) process

From the definition we know that a stochastic process $(X_t)_{t\in\mathbb{Z}}$ is called strictly stationary if for all $h\in \mathbb{N}$ the distribution of $(X_t,X_{t+1},\ldots,X_{t+h})$ is ...
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2answers
44 views

formal proof $\mathbb E[g_2(Y) \cdot g_1(X)|Y]=g_2(Y) \cdot \mathbb E[g_1(X)|Y]$

I am looking for a formal proof of the following: $\mathbb E[g_2(Y) \cdot g_1(X)|Y]=g_2(Y) \cdot \mathbb E[g_1(X)|Y]$ where $X$ and $Y$ are continuous random variables and $g_1, g_2$ are some ...
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1answer
50 views

Is this a normal distribution?

Given $Z\sim N(0,1)$, a coin is flipped. Another random variable $$ W = \begin{cases} \phantom{-}Z & \text{if heads}, \\ -Z & \text{if tails}. \end{cases} $$ Is $W$ normal? I just am not ...
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0answers
26 views

Probability of going from a set $S$ to its complement on a Markov chain

I need to show that if $\pi$ is the stationary distribution of a Markov chain $M$, then for every set of vertices $S$, the probability to choose a random node in $S$ according to $\pi$ and then going ...
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1answer
15 views

Probability and statistics(numbers inside a bowl)

So, inside a bowl there are numbers from 0 to 9 (0,1,2....,9). If you take out 3 random numbers without putting the number back after each extraction, and compose a number by those 3 digits in ...
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0answers
34 views

Show that $P(\limsup A_n)=1$ if for each $k$ the series $\sum_{n>k} P(A_n|A_k^c ..A_{n-1}^c )$ diverges

Show that $P(\limsup A_n)=1$ if for each $k$ the series $\sum_{n>k} P(A_n|A_k^c ..A_{n-1}^c )$ diverges. Can we say that the second Borel-Cantelli lemma follows directly from this?
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20 views

Stratonovich integral of Wienere process [duplicate]

I need an help with the following exercise. Let $(W_t)_{t\geq 0}$ a Wiener process on $(\Omega, \mathcal E, \mathbb P)$ and let $I=[0,T]$ be an interval. We want to prove that the Stratonovich ...
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1answer
85 views

Expected Value of Mixture Distribution

I have never before encountered a "mixture distribution," so I have run into a little trouble trying to calculate the mean of this one: Let $X_{a,b}$ be such that, for parameters $a \in (0,1)$ and $b ...
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1answer
105 views

Studying the probability of an event with a continuous distribution

Let $W:=(W_1,W_2,W_3,...,W_k)$ be a random vector of dimension $k \times 1$ where each $W_j$ has a continuous uniform distribution in $[a,b]$, $0<a<b$. Let $1\{.\}$ be an indicator function ...
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1answer
40 views

The Continuity of Correlation Coefficient of a Continuous Stochastic Process

Given a continuous stochastic process with respect to time with finite variance at a given $t$. Does it necessarily imply, as $d\to 0$, 1) the covariance between $t$ and $t+d$ approaches the variance ...
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1answer
27 views

In General what is the conditional density of Y given X=i when Y is continuous and X is discrete.

I need help understanding why it would be the case that $$f_{Y\mid X}(y\mid i)=\frac{P(X=i\mid Y=y)f_Y(y)}{P(X=i)}.$$ Though I've just started studying conditional distributions, I am comfortable ...
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1answer
30 views

Probability of getting disease and Markov chain

I am studying marcov chain. The question is . There are 5 people ( 4 diseased / 1 healthy) Two people are selected randomly and assumed to interact. If one is diseased and the other is healthy, ...
3
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1answer
56 views

Show that $E[(Y-E[Y|X])*(E[Y|X]-g(X))]=0$

$g$ is a measurable function and $X$ and $Y$ are continuous random variables, we need to show that: $E[(Y-E[Y|X])*(E[Y|X]-g(X))]=0$ My attempt: ...
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0answers
25 views

Does anybody know of the proof of conditional entropy in general

I was wondering if someone could provide a proof of conditional entropy in general or with two and three variables or a place where I could find it. I am having trouble with some of the algebra and ...
2
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1answer
29 views

Old qualifier problem clarification (Probability related)

I'm trying to make sense of what is being asked in the question. What does the set $E_{n}^{\epsilon}$ represent? Let $\mathbb{P}$ be a probability measure on $\mathcal{B}(\mathbb{R})$. Let ...
2
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0answers
36 views

Almost sure convergence and boundedness

Let $(\Omega, \mathscr{F}, \mathbb P)$ be a probability space and $(X_n)_{n \in \mathbb N}$ a sequence of random variables on that space, which converge almost surely to a constant $c\in \mathbb R$, ...
2
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2answers
126 views

Product of stochastic integral and brownian motion

I am trying to compute the following expectation: $$ M_T = \mathbb E\left[W_T\int_0^T\,t\,d W_t \right] $$ where $0<t<T$ and $W = (W_t)_{t\geq 0}$ is a standard Brownian Motion started at $0$. ...
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1answer
85 views

Studying the probability of an event

Let $W:=(W_1,W_2,W_3,...,W_k)$ be a random vector of dimension $k \times 1$ taking values $(1,0,0,..,0)$, $(0,1,0,...,0)$,$(0,0,...,1)$. Assume that $W$ has a discrete uniform distribution. Let ...
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1answer
58 views

Special case of variance decomposion formula

The end of the preamble of the wikipedia page for the law of total variance provides the following formula for the variance of $X$ where $A_1,A_2,\ldots,A_n$ is the partition of the outcome space ...
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0answers
51 views

Sub sigma-algebra Example

I'm looking at the sub $\sigma$-algebra example on Wikipedia, and I don't understand the notation that is used. The example defines the $\sigma$-algebra $G_n = \{ A \times \{H,T\}^{\infty}\ : A ...
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20 views

Gaussian Free Field and Branching Random walk

I was told that there is a duality between gaussian free field in 2-D and branching random walk. Could anybody please give a brief explanation how both probability theory are dual to each other? ...
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1answer
23 views

Where do the forumlas for expectation and variance for geometric and Poisson distributions come from?

Okay so I have been given a list of 4 distributions and their respective mean(expected) and variance. I can see where the Bernoulli and Binomial ones come from using the definition of expectation and ...
2
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2answers
106 views

Bounding profits of gambler by Azuma Inequality

A gambler plays the following game: In each round, he can pay any $0 < p < 1$ dollars, and win 1 dollar with probability p (independently). Show that the probability that the gambler's net ...
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0answers
72 views

An example of switching expectation and infimum

Suppose $(X_t : t \geq 0)$ is a real valued stochastic process. In my case, it takes values in $[0,1]$, starts at $1$, and is decreasing. (It is also Markov). I have calculated $\mathbb{E} (X_t) =: ...