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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3
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1answer
78 views

If $X_n \rightarrow X$ almost surely then $f(X_n) \rightarrow f(X)$ almost surely

Proof: If f is continuous and $X_n \rightarrow X$ almost surely, then $f(X_n) \rightarrow f(X)$ almost surely. Thats the only information I have. Does this only hold if the measure on the target ...
3
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0answers
36 views

Equality of conditional expectations [duplicate]

Let $X,Y$ be integrable random variables such that $\mathbb E(X|\sigma(Y))=Y$ and $\mathbb E (Y|\sigma(X))=X$, where $\sigma(X)$ is the smallest sigma algebra such that $X$ is measurable. Show that ...
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1answer
158 views

Between bayesian and measure theoretic approaches

I was wondering how a bayesian statistician would approach the problem of defining a probability density function for a random variable. In a measure theoretic sense, If the distribution of the ...
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2answers
62 views

Condition on moment generating functions implies independence

Let X, Y be random variables so that $M_X(a) = E[e^{aX}]$ and $M_Y(b) = E[e^{bY}]$ are finite for all $|a|, |b| < \delta$ for some $\delta > 0$, and $E[e^{aX + bY}] = E[e^{aX}]E[e^{bY}]$ for ...
1
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1answer
125 views

Brownian Motion inequality (related to Dvoretzky-Erdoes test)

i have the following question: Let $B(t)$ be a d-dimeansional Brownian motion $d\ge 3$, and $f$ be a monoton increasing function from the positive reals to the positive reals. Let $A_n=(\exists t\in ...
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2answers
101 views

Sub-Martingale and Martingale

An integrable sub-martingale $S_t$ with $\mathbb E(S_t)$ being a constant is a martingale. Is this statement true, please? I think so.
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1answer
33 views

Brownian Motion and Progressive Process

Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the ...
0
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1answer
39 views

How does conditional expectation work when the conditional value isn't given?

For example, if we have the following random variables: $Y\sim Bern\left(\frac{1}{5}\right),$ $ Z = \left\{ \begin{array}{l l} X & \quad \text{Y=1}\\ -4X & \quad \text{Y=0} ...
1
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1answer
44 views

Probability of Getting a Rational Number

Since for a continuous random variable lets say, $X\sim N(0,1)$, we know that $P(X=x)=0$ for any $x\in\mathbb{R}$. I was wondering since we can extend probability measures over countable unions that ...
3
votes
1answer
763 views

Best Rigorous Probability Theory Textbook 'without' Measure Theory?

This question follows from my previous question. I want a book that deals probability theory rigorously (and cover as many topics as possible) yet not involving much about measure theory. There are ...
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3answers
138 views

Extremely ($90$%) biased coins. What information can we derive/assume based on results of only $10$ coin flips?

Let's suppose there are $2$ heavily biased coins such that coin A has a bias of coming up $90$% heads and coin B has a bias of coming up $90$% tails. Both coins are placed in a bag and one is ...
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0answers
39 views

counterxample: Events with bounded minimum probability not having a subsequence with 0 probability

So, I'm trying to come up with a counterexample (Or proof if that would be easier that thi thing is false, but it seems like counterexample is the normal way to show something is false...) Anyhow, ...
2
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1answer
305 views

Analogous of Markov's inequality for the lower bound

Consider a positive random variable $X$ and call $E[X]$ its expectation. For any positive $a \in \mathbb{R}$, an upper bound for the probability of $P(X>a)$ is provided by the Markov's Inequality, ...
3
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1answer
1k views

On clarifying the relationship between distribution functions in measure theory and probability theory

I recently found myself confusing concepts from measure theory and probability theory, so I'd like to get an idea for what I'm misunderstanding. This definition is what started it all: A sequence ...
0
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1answer
43 views

Stochastic Processes, requirement at ''source" probability space, is it always an product over $T$?

Let $(\Omega, \mathcal F, P)$ be a probability space, and let $(S, \mathcal S)$ a set $S$ together with a $\sigma$-Algebra over $S$, also let $T$ be some index set, then for each $t \in T$ let $X_t : ...
2
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0answers
31 views

probability of clusters for iid points

Consider that $X_1^{(n)},...,X_n^{(n)}$ are iid uniform random variables on $[0,n]$. For $T >0$, let $N_n(T) = \sup_{t \in [0,n]} \# \{ i: |X_i^{(n)} - t| \leq T \}$ be the maximum number of points ...
0
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1answer
89 views

Query relating to marginal pdf (probability density function)

I have a set of two related queries relating to marginal pdfs: i.How to proceed finding the marginal pdfs of two independent gamma distributions (X1 and X2) with parameters (α1,β) and (α2,β) ...
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0answers
31 views

Probability question on 5 components operation after a given time

My attempt: $\lambda=1/2.5=0.4$ Since $P(T\geq t)=({1-e^{-\lambda t}})$ $P(T\geq3)=({1-e^{-0.4(3)}})^5$ However my book says: $P(T\geq3)=({e^{-0.4(3)}})^5$ Why is this? How did the book do ...
2
votes
1answer
113 views

How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by ...
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0answers
42 views

Convergence of a distribution function

I have a problem that has me stumped. Frustratingly enough I can't really get the problem started. It goes as follows: Let $X_{1},X_{2},...$ be independent $C(0,1)$-distributed random variables. ...
2
votes
1answer
63 views

Is $\mathbb E(X|\mathcal G)$ an integral of $X$ with respect to some measure?

My instructor defined $\mathbb E(X|\mathcal G)$ in the usual way and mentioned that it can also be characterized as an integral of $X$ with respect to some measure. Similarly to $\mathbb E(X|A)=\int X ...
2
votes
1answer
277 views

Progressively Measurable for Rigth Continuous Adapted Processes

Any adapted and right continuous process $X_t$ is progressively measurable. For the above statement, I found proof in several books. They all have similar argument as follows. For a given $t > ...
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0answers
59 views

Integral of a standard brownian motion

I am working on the following problem which is on an introductory chapter of Brownian motion: Let B(t) be the standard Brownian motion. Define $X(t)=[1/\sqrt{t}]\int_{0 \to t}{f(B(s))}ds$ where $f$ ...
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0answers
38 views

Independent events satisfying $P(A_n)<1$ [duplicate]

So I'm working on my mathematical probability class, and I have the following question: Suppose $\{A_n\}$ are independent events satisfying $P(A_n)<1$ for all $n$. Show $P(\bigcup _{n=1}^\infty ...
0
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1answer
39 views

Exponential limit on sum of probabilities guarantees the product of powers of expectations is integrable

If X, Y are random variables and there exists a constant $c>0$ so that $P(|X| \geq x) + P(|Y| \geq x) \leq e^{-cx}$ for all x > 0, then $E[X^m Y^n]$ is integrable for all nonnegative integers m, ...
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0answers
37 views

How to find union and intersection of events?

I have sample space of experiment $S=\left\{x|-\infty<x<\infty\right\}$. I consider events $$A_i=\left\{x \;\middle|\;\frac{1}{2^{i-1}}\le x<\frac{3}{2^i}\right\};i=1,2...$$ And I want to ...
1
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1answer
75 views

Find conditional probability $\mathbb{P}(X \le x | \max(X,Y)) $

Let $X,Y$ be iid such that $X\sim F>0$ and $Y \sim F>0$ ($X$ and $Y$ have the same probability distribution). Find $\mathbb{P}(X \le x | \max(X,Y)) $. I know that $\max(X,Y) \sim F^2$. I ...
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0answers
13 views

Proving $Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{-1}{n_i-1}$ for $ j \neq k$

For the model of a single factor experiment: $y_{ij}= \mu + \alpha_i + e_{ij}$, $(1 \leq i \leq a, 1 \leq j \leq n_i)$, where a = the number of treatments, $n_i$ = the number of experimental units ...
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0answers
27 views

Principal Components vs Principal Directions

I'm trying to do statistical downscaling of some climate data and there is a module of principal component analysis by regression method required. I am confused with the different terms here. What is ...
1
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1answer
88 views

How does a Nakagami Random Variable behave?

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ I have two questions regarding this random variable, 1- Is a sum of ...
0
votes
1answer
143 views

how many types of events actually exists in the theory of probability?

I read many article on the internet and found that there are only three types of event that can be occurred(or that has been considered in the probability theory). those are : mutually exclusive ...
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2answers
233 views

Prove X is a martingale

Prove $X = (X_n)_{n \geq 0}$ is a martingale w/rt $\mathscr{F}$ where X is given by: $X_0 = 1$ and for $n \geq 1$ $X_{n+1} = 2X_n$ w/ prob 1/2 $X_{n+1} = 0$ w/ prob 1/2 and $\mathscr{F_n} = ...
0
votes
1answer
975 views

Coin-flipping experiment: the expected number of flips that land on heads

This question is from Sheldon M. Ross: Introduction to Probability Models which is about finding the expectation by conditioning. Question: A coin, having probability $p$ of landing heads, is ...
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0answers
76 views

Random walks: number of crosses between $-\sqrt{x}$ and $\sqrt{x}$

Let $S_n = \sum_{k=1}^n X_i$ be a simple random walk, where $X_1, X_2, \dots$ are independent Bernoulli random variables, $\mathbb{P}(X_k = 1) = \mathbb{P}(X_k = -1) = \frac 1 2$. Let $T_1 = 1, ...
0
votes
1answer
60 views

Brownian Bridge equivalence of definitions

How can I verify the equality of the distributions arising from the two definitions of Brownian Bridge here: http://en.wikipedia.org/wiki/Brownian_bridge The two definitions are $W_t-tW_1$ and ...
0
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2answers
57 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
0
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1answer
47 views

A question on semi-martingale and its variations

With probability one, paths of semimartingales have unbounded variation. What I know is that a martingale is also a semi-martingale, for example, Brownian motion. Hence, Brownian is an example of ...
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2answers
52 views

What is wrong with my method of finding the probability

One way to solve this and my book has done it is by : This is a well known way, but I have a different method, and it seems logical to me (but I don't know what the mistake is). And yes it's ...
0
votes
1answer
72 views

Does $x \perp (y,z)$ imply $x \perp y \mid z$?

Does $x \perp (y,z)$ imply $x \perp y \mid z$, where $\perp$ denotes stochastic independence? I was told it is true and the following is the proof (which I believe is wrong): We want to show that ...
3
votes
1answer
65 views

General Weak Law of Large numbers

I came across a question regarding the WLLN. Suppose for $X \geq 0$ , $\mathbb{E}[X] = \infty $ , $S_n = \sum_{i \leq n} X_i$, $X_i$ are iid copies of $X$ , and $\frac{\mathbb{E}[X \mathbf{1} _{X ...
3
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0answers
50 views

Does $(X \perp Y\mid Z) \ \& \ (X \perp W \mid Z) \implies X \perp (Y,W) \mid Z$ hold?

I know that $ X \perp (Y,W) \mid Z \implies (X \perp Y\mid Z) \ \& \ (X \perp W \mid Z)$ but does the converse hold? i.e. does: $$(X \perp Y\mid Z) \ \& \ (X \perp W \mid Z) \implies X \perp ...
2
votes
2answers
194 views

Zero mean but not a martingale

I am looking for a simple stochastic process which has zero mean for all $t\geq0$ but it is not a martingale. I been looking in to local martingales but having trouble keeping the mean zero for all ...
0
votes
1answer
34 views

Martingale property of negative Brownian motion

Let $B_t$ be Brownian motion, with $B_0=0$. Next define $M_t=-B_t$. Have I understood it correctly that $M_t$ is not a Martingale? $E[M_t]=0$ $E[M_{t+1}|M_t]=-M_t$ and therefore not a Martingale? ...
0
votes
2answers
45 views

Is this true: $\lim_{\lambda \rightarrow 0}E\left[ e^{-\lambda X} \right] = P\left(X < \infty \right)$?

Is it true that $\lim_{\lambda \rightarrow 0} E\left[ e^{-\lambda X} \right] = P\left(X < \infty \right)$ ? I've seen this in a few places online but I can't seem to be able to find a proof online ...
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0answers
36 views

Subgaussian bounds for $X$ imply subgaussian bounds for $X-E(X)$

A random variable $X$ is called sub-gaussian, if there exist positive constants $C,c$ such that for all positive $\lambda$ we have $$P(|X|\geq \lambda)\leq Ce^{-c\lambda^2}.$$Now I'm reading a text, ...
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votes
1answer
162 views

A Question about an example in Durrett's Probability textbook

I was reading an example in Durrett's book: Probability : Theory and Example, 4th edition (pdf verison) (Example 3.4.7, p.112) The scenario is as follows: Define $Y_1,Y_2...$ be independent ...
0
votes
1answer
31 views

Jump Set v. Range of Randome Variable

What is the difference between the range of a random variable X, and its jump set? I know that they are not equivalent sets, e.g. for a continuous RV, the range is $(- \infty , \infty)$, but the jump ...
4
votes
1answer
221 views

Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
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2answers
69 views

Distribution of a distance between random numbers

I'm working on a problem in which I came to a question concerning distribution law of a result of operations on random variables. I will ask a simple question and hope to understand the concept from ...
1
vote
2answers
111 views

What is the sum-capacity for a non-symmetric interference channel for information theorists?

This question is dedicated for people who are experts in information theory. An interesting result for a two user interference channel in information theory, is the sum-capacity to within one bit. It ...