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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22 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 ...
0
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1answer
96 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 ...
-1
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1answer
115 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 ...
1
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1answer
45 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 ...
0
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1answer
29 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
39 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
67 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: ...
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
54 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
210 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
84 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
86 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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1answer
30 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
113 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 ...
1
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0answers
92 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) =: ...
0
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1answer
46 views

Find a sequence of events $A_n$ for which all three inequalities…

Let $(\Omega,F,P)$ denote the probability triple for the discrete uniform distribution on the set $\{1,2,3,4\}$. Find a sequence of events $\{A_n\}$ for which these inequalities hold: (i) For each ...
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3answers
386 views

Why does randomness exhibit a pattern in the long run?

!!! Layman here so please avoid complex math and answers. Random (usually pseudorandom) events are usually characterized along these lines: Each outcome in a trial experiment must be i.i.d.; i.e. ...
1
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1answer
28 views

Finding measure given constant margins

Suppose $g:[0,1]^2\to R$ and $g$ can have finitely many discontinuities. $F$ is continuous and atomless c.d.f on $[0,1]$ $$\int_{[0,1]} g(x,y)dF(y)=1/2, \forall x$$ $$\int_{[0,1]} g(x,y)dF(x)=1/2, ...
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0answers
52 views

A question about weak convergence of random variables

I am reading my lecture notes and our definition of weak convergence or random variables is: First another definition: A sequence $\mu_n$ of probability measures on $\mathbb R$ converges weakly to a ...
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1answer
77 views

Continuity of the Second Moment 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 the second moment and variance are continuous in time?
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1answer
38 views

Convergence of random variables: struggle with a proof

I am trying to understand a proof of the following theorem: $X_n$ is a sequence of random variables. $X_n \to X$ in probability $\implies$ that each sub-sequence of $X_n$ has a sub-sequence which ...
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1answer
57 views

Fingerprinting and randomized algorithms

My question is regarding the notes pages 1-2 specifically http://www.cs.berkeley.edu/~sinclair/cs271/n3.pdf I understand everything up to the point near the top of the second page where it says ...
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1answer
35 views

Card shuffling transition matrix

a short understanding question. Consider a pile of $n$ cards. At every step we choose randomly 2 cards and transpose them. Now $X_n$ should be a Markov chain which describes the order of the pile at ...
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1answer
102 views

Determining a transition probability matrix

If I have that $X_n$ is a two-state Markov chain whose transition probability matrix is: $P = \left( \begin{smallmatrix} \alpha & 1-\alpha\\ 1-\beta & \beta \\\end{smallmatrix} \right)$ ...
1
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1answer
37 views

tail inequality for expectations

I would like to upper bound the expectation $$ \mathbb{E}[X \, \textbf{1}\{X > t\}], $$ where $\textbf{I}\{p\}$ evaluates to $1$ if $p$ is true, $0$ otherwise, and $X$ is some non-negative random ...
6
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2answers
135 views

Approximation of conditional expectation

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. Let $\mathcal{A}$ be a complete sub-$\sigma$-algebra of $\mathcal{F}$. For the moment assume that $X$ is a random variable with ...
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0answers
51 views

Proof that limit exists in $L^2$ sence

Proof that exists $L^2$ limit $$ \lim_{\varepsilon\downarrow 0} L(t,\varepsilon)=\lim_{\varepsilon\downarrow 0}\frac{1}{\varepsilon}\int_0^t\mathbf{1}\left(W_s\in(-\varepsilon,\varepsilon)\right)ds, ...
0
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1answer
60 views

Joint Probability and Intersection Probability

Given two independent events A and B: $P(A \cap B)= P(A)*P(B)$ but then I saw somewhere that: $P(A \cap B)= P(A)*P(B)= P(A|B)*P(B) = P(B|A)*P(A)$ where for example $A$ is $X=x$ and $B$ is $Y=y$ ...
2
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1answer
65 views

Elementary event in event space

I encountered a very basic question of probability. Consider the sample space Ω = {a,b,c,d} and assume that the only elementary events in the Event space F defined on Ω are {a} and {b}. Explicitly ...
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1answer
49 views

How to find $\mathbb P(XY<\frac{1}{2})$ and $\mathbb P(Y< X^2)$ without convolution?

Let $X$ and $Y$ be uniformly distributed on $[0,1]$ and independent random variables. Find $$\mathbb P\left(XY<\frac{1}{2}\right) \text{ and } \mathbb P\left(Y< X^2 \right).$$ Tip: one can do ...
1
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1answer
31 views

Independent identical distributions and conditions for equality

Suppose $X,Y,Z$ are independent identical distribution, taking values in a finite set $x_1,...,x_n$. Is the following true? $$Pr\{X=Y: Y\neq Z\}\leq Pr\{X=Y\}$$ What do think about the condition for ...
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2answers
49 views

Proving certain aspects of Entropy

I am trying to prove three properties of entropy. $1)$ $H(X|Y,Z)\le H(X|Y)$ $2)$ $H(X|Y,Z)\le H(X,Y)$ $3)$ $H(X,Y,Z)+H(Y)\le H(X,Y)+H(Y,Z)$ I have proved the third one, but it is based on part 1. ...
2
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1answer
48 views

Series of moments of random variables

I am interested in the convergence of the series $$ \sum_{n=1}^\infty\frac1n\operatorname E[|X|^pI_{\{|X|>b_n\}}], $$ where $X$ is a random variable with $\operatorname E|X|^p<\infty$ for some ...
0
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1answer
36 views

Describe the push forward of the borel probability measure on $\mathbb{R}$ under its Cumulative distribution function

Will the push forward be the borel probability measure on [0,1]. If so, how to show this? Kindly help
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0answers
184 views

Prove that a process has independent increments

Let $\tau_1,\tau_2,\dots$ be a sequence of i.i.d. random variables such that $\tau_i \sim Exp(\lambda)$ on $(\Omega,\mathcal E,\mathbb P)$. Then define $T_i=\sum_{k=1}^i \tau_k$, so we know how each ...
3
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0answers
67 views

An irreducible Markov chain is a martingale

Let $\{X_n\}$ be an irreducible Markov chain. Does exist example of such $\{X_n\}$ which is also a martingale given that: a. $\{X_n\}$ is recurrent with finite number of states (but bigger ...
2
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0answers
42 views

If $E(|X|\log|X|)<\infty$ then is $E\left[\frac{|S_n|}{n}\ \log\left(\frac{|S_n|}{n}\right)\right]<\infty$?

I am trying to finish a homework problem in my probability class. I think I am at the end of my problem if I can show that $$E(|X|\log|X|)<\infty$$ implies that $$E\left[\frac{|S_n|}{n}\ ...
2
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0answers
21 views

Proving the sample variance has a chi squared critical value

Let $X_1, . . . , X_n$ be independent normal observations with means $µ = 0$ and variances $σ^2$. For testing the null hypothesis $H_0 : σ^2 = 1$ versus the alternative $H_a : σ^2 > 1$ show that ...
1
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2answers
64 views

Continuity of $\mu \mapsto \mu(E)$ for $\mu$ probability measure and $E$ Borel subset

Let $X$ be a topological space endowed with the Borel sigma-algebra, let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, endowed with the weak* topology. Fix $E$ Borel subset of ...
0
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1answer
8 views

If Y and X are ind. binomial RV with parameters (n, p) and (a, b) respectively, then (Y/n) - (X/a) is approximately distributed. Find V(Y/n - X/a).

I tried to find E(Y/n - X/a) and said it was E(Y/n) - E(X/a)= p - b. But then I got stuck finding the variance, I wasn't sure if it needed to be done with moment generating functions or if the Central ...
2
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1answer
72 views

Show that if the sum of an diverges, no discrete probability space can contain independent events

Suppose that $0\leq p_n\leq 1$, and put $a_n= \min \{p_n, 1-p_n\}$. Show that if $\sum a_n$ diverges, then no discrete probability space can contain independent events $A_1, A_2, \ldots$ such that ...
2
votes
2answers
444 views

Find three events that are dependent but pairwise independent

Let $(\Omega, \mathcal F, P)$ denote the probability triple for the discrete uniform distribution on the set $\{1,2,3,4\}$. Q. Give an example of three dependent events with probabilities strictly ...
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1answer
63 views

Prove that the following is a field. [closed]

Let $\mathcal{F}_1,\mathcal{F}_2,...$ be classes of sets in a common space $\Omega$. (a) Suppose that $\mathcal{F}_n$ are fields and that $\mathcal{F}_n\subset\mathcal{F}_{n+1}$. Show that ...
2
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1answer
153 views

Does this argument suffice to show a “record” occurs at time n with probability 1/n?

I think it does, but, in addition to checking for correctness, I'd like to know what other argument we might use. Let $X_1, X_2,...X_n$ be be a sequence of independent identically distributed ...
2
votes
1answer
68 views

$X_n \stackrel{d}{\to}X$, $Y_n \stackrel{d}{\to} c \implies X_n+Y_n \stackrel{d}{\to} X+c$

Let $X_n\Rightarrow X$ and $Y_n\Rightarrow c$. Show that $X_n+Y_n\Rightarrow X+c$. Prove: There exists sequences of random variables $(X^{(*)}_n)$ and $(Y^{(*)}_n)$ such that $(X^{(*)}_n)$ and ...
2
votes
1answer
56 views

Fatou for weak convergence

I want to do exercise 3.2.4 from Rick Durett, Probability: Theory and Examples page 86. $$\text{Let } g\geq0 \text{ be continuous. If }X_n \Rightarrow X_{\infty} \text{ then } \liminf_{n\rightarrow ...
1
vote
2answers
130 views

Expected first return time of Markov Chain

Given the following Markov Chain: $$M = \left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{4} & \frac{3}{4} & 0 & 0 & 0 & 0 ...
3
votes
1answer
43 views

Writing random variable formulas with set notations, What is the problem?

Is it wrong to write $\displaystyle P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$ when $X$ and $Y$ are random variables? As I know a random variable is a function and therefore has a range and the two ...
3
votes
1answer
46 views

Prove that if $E(X\log X)<\infty$ then $E(\sup_n |S_n|/n)<\infty$.

This is part 2 of a two part question. In the first part, we were asked to show that if you had a non-negative sub martingale $M_n$ then $$\sup_n E(\sup_{k\leq n} M_k)\leq \sup_n 2E(M_n \log M_n)+2$$ ...