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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Quantifying over all random variables

I often encounter statements in the literature in probability theory of the form: "Let $(\Omega, \mathscr{A}, P)$ be a probability space, $S$ a state space and $X : \Omega \to S$ a random variable ...
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how to prove $\mathop {\lim }\limits_{n \to \infty } {\{\Phi [(1 - \varepsilon )\sqrt {2\log n} ]\}^n}=0$?

$\Phi (x)$ is the distribution function of standard normal distribution. $\varepsilon$ is some positive tiny number that is less than 1. How to prove this beautiful and important limitation: ...
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64 views

IId random variables from Exponential distribution

If $X_1$ and $X_2$ are iid random variables from the exponential distribution with parameter $\lambda$. I need to find the pdf of $X_1/(X_1 + X_2)$. As of now I have used $X_1=\lambda e^{-\lambda x}$ ...
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66 views

Cdf and Pdf of independent random variables(iid)

Let $X_1, X_2,...,X_n$ be independent random variables, each having a uniform distribution over $(0,1)$. Let $Z:=\min(X_1, X_2,...,X_n)$ and $Y:=(X_1, X_2,...,X_n)$. I need to find the cdf and pdf of ...
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64 views

Itô formula + SDE

I have a problem with solving the following problem: I.e. I want to show that $X_t$ is a solution to the SDE by employing the Itō formula. Now the problem is I don't get how I should set the ...
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1answer
50 views

Lebesgue-Stieltjes Integral (Several Variables)

Let $\mathcal F$ be a convex set of probability measures or distribution functions and $F, G$ be two elements in $\mathcal F$. Let $T$ be a functional on $\mathcal F$ defined as follows. Note that $h$ ...
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2answers
76 views

Proof of the law of large numbers for higher moments

Let us work on some probability space $<\Omega,\mathscr{A},\mathbb{P}>$: I'm looking for (independent) proofs of two proofs, of the generalised weak and strong law of large numbers ...
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1answer
61 views

Counterexample to conditional probability with dependent events

Let $X1,X2,X3$ be i.i.d. taking values in a finite set, and not constant. Is it necessarily true that $P(X3=X2|X2≠X1)≤P(X3=X2)$? Give a proof or a counterexample. Since the two events $A=\{X3=X2\}$ ...
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18 views

Is it true that for every signed probability distribution `f`, there are positive distributions `g` and `h` st. `fg=h`?

While reading the article Half of a Coin: Negative Probabilities, I came across the following theorem: For every generalized g.f. f (of a signed probability distribution) there exist two ...
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1answer
22 views

Fnd a sequence to be convergence in distribution

Varablies $X_1,\ldots ,X_n$ are independent and $\forall {i\in\{1,\ldots n\}}: X_i \sim \exp(1)$. Find numeric string $a_n$ such that sequence of random variables $$Y_n= \max\{X_1,\ldots, X_n\} - ...
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27 views

How to find relation between expected value and P(X>0)

Let random variable Y ≥ 0 with E[Y]^2 < ∞. Prove that P(Y > 0) ≥ (E[Y])^2/E[Y^2] ? How will we approach this question
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35 views

Simple question about an equality of a stopped process

Let $T$ be stopping time, and $X_n$ be stochastic process. Then the stopped process $X_{n \wedge T}$ can be written as $$X_{n \wedge T} = X_n 1_{T \ge n} + X_T 1_{T < n}$$ where $1_{(\cdot)}$ is ...
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17 views

Potentials and Markov Processes

Given a resistive electrical circuit $G$, i.e. a graph with nonzero weights attached to each edge whose reciprocal we call the "resistance," we can define a reversible markov chain on the graph, ...
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2answers
120 views

Proof of a formula for the expectation of a product of random variables

I want to prove the second task, task b) (see picture below). a) was not hard to show. One question before I start: I am a bit confused about the notation, but $\mathbb 1(t)_{\{Y>t\}}$ is $1$ if ...
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2answers
68 views

Is there any “randomness” in a random variable?

I've been using probability theory (, statistics, bayesian inference) for a while - and I find it very useful and mathematically elegant, but I still can't get where is the hidden "randomness" in a ...
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1answer
41 views

Example of convergence in probability to a non-degenerate rv

Suppose the sequence of random variables, X$_n$, converges in probability to another random variable X. The condition requires that for any arbitrary distance, $\epsilon$, the probability that the ...
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33 views

An application of Strassen's theorem

Recently I handed in a problem set containing the following question, but neither myself nor my classmates managed to find a satisfying solution. We were quite certain that a fruitful approach was to ...
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26 views

iid random variables and stopping time

This is Exercise 14.30 from Probability for Statistics and Machine Learning. Let $X_i$ be iid with $E|X| < \infty$, and let $T$ be a stopping time adapted to $\{ X_i \}$. Let $S_n = ...
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1answer
106 views

the meaning of bound of characteristic function in the neighborhood of zero

Let $\{X_n: n=1,2,\ldots\}$ be a sequence of integrable random variables. Let $\{\phi_n: n=1,2,\ldots\}$ be the corresponding characteristic functions. Suppose that we have $$ |1-\phi_n(t)|\leq A ...
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1answer
24 views

almost sure convergence of iid sum

I'm trying to reconcile what's intuitive with almost sure convergence. Say I have iid $X_i$ with $P(X=+1) = \epsilon$, and $P(X=-M) = 1-\epsilon$, where $M$ is some very large (say $10^{100}$) number ...
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2answers
47 views

Marginally continuous measures

Consider a continuous density f on $\mathbb{R}^{2}$, and suppose that $\mu$ is the corresponding Lebesgue-Stieltjes measure on the product space ...
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1answer
36 views

Expression for characteristic function of a truncated RV

Let $\langle\Omega,\mathscr{F},\mathbb{P}\rangle$ be a probability space, let $X$ be a random variable defined thereon with density $f$ and $\phi$ be its characteristic function. Then if $A \in ...
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1answer
30 views

Let $X \in [a, b]$ be a random variable. Is it true that $F_{X}$ is strictly increasing in $[a, b]$ iff $X$ is continuous?

Inspired by distribution of (inverse) distribution function. This made me think: is the following statement true? Let $X$ be a random variable with support in $[a, b]$. $F_{X}$, the cumulative ...
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28 views

Expression for joint probability

I have two expressions for $P(X_i|y=0)$ and $P(X_i|y=1)$, where each expression is a multinomial distribution and $y\in \{0,1\}$. I'm interested in finding the joint log likelihood, and thus I'm ...
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3answers
206 views

How to work with series of uncountable objects

Let $A_\beta$, $\beta \in B$ be a family of pairwise disjoint events. Show that if $P(A_\beta) > 0 \ \ \forall \beta \in B$, then $B$ must be countable. My work Suppose $B$ is ...
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1answer
43 views

Markov chain modes of convergence

This is continuation of the question stated here. Let $\left( {{X_\alpha }:\alpha \in A} \right)$ be a finite space Markov chain (discrete or continuous), consisting of only transient and absorbing ...
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1answer
47 views

I'm confusing with inequality of p-norm

As far as I know, for $p \ge 1$, $||X||_p \equiv (E|X|^p)^{1/p}$ becomes a norm in probability space. If this is right, those two inequalities on each link seem to contradict with each other. ...
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30 views

iid sums of zero mean rv drops below zero almost surely

Let $S_n = X_1 + ... + X_n$, where $X_i$ is iid with $E[X_i] = 0$ and $E[X_i^2] < \infty$. I am interested if it is necessarily the case that $$ P(\{ \omega : S_n(\omega) < 0 \text{ for some } ...
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44 views

Monotone convergence theorem allows the limit to be infinity

Monotone convergence theorem(MCT) doesn't impose any restriction on the limit. For example, if $\{X_n\}$ satisfies $0 \le X_n \nearrow X$ with $EX=\infty$, then I still could use MCT to get ...
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51 views

Some properties of a random variable

I have absolutely no idea how to show this: Let $X$ be a random variable whose distribution is not degenerate. By considering the function $F( \theta) = \mathbb{E} U( \theta X)$, $\theta \in ...
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1answer
33 views

BSC channel probability, (binary symmetric channel)

I have question regarding the binary symmetric channel (BSC), which assume each channel use is indepedent (i.e, if you send a '0', then you send '1', each time you send it is indepedent of others). ...
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28 views

Probability with statistics using 'R'

I've been going through a few previous exam questions and came across this one and computed in the programing 'R'. ...
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1answer
107 views

perfectly correlated processes

I am really stuck in this question: Let $\{S_t\}$ and $\{S'_t\}$ be two stochastic processes, satisfying \begin{equation} dS_t = S_t ( \sigma_t \,dB_t + r_t \,dt), \quad dS'_t = S'_t (\sigma'_t ...
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1answer
38 views

If the expected value is on the boundary of the range, then the random variable is a.s. constant

Let $X$ be a real-valued random variable on $\Omega$, $I\subseteq\mathbb{R}$ be an interval, $X(\Omega)\subseteq I$ and $E[|X|]<\infty$. Why does $E[X]\in\partial I$ imply that $X=E[X]$ almost ...
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27 views

Peak absolute variation of a Band-Limited Process around its current sample for a given horizon T

Is it straightforward to find a bound on the maximum possible absolute variation around the mean or the last sample of a band-limited process for a given time horizon like $T$? More specifically, how ...
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19 views

Expected Value and Variance

i'm just breaking my head dealing with this question. suppose we toss a coin 1000 times independently, let X be the number of sequences of 7 times "head". with probability p for head. what is the ...
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2answers
80 views

Stroeker Problem: Sum of consecutive cubes being a perfect square

I encountered to following textbook problem in the book 'Introduction to probability' (p.34) by Blitzstein and Nwang. NO homework, but self-study ! Part a) is no problem, but b) struck me down. ...
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Is set of product distributions compact under second moment constrains?

Please do not treat this question as duplicate of Definition of the set of independent r.v. with second moment contstraint which I didn't want to edit because of many useful comments. Also, in this ...
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1answer
319 views

Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a ...
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1answer
27 views

An Application of law of large numbers.

Let {$X_k$} be a sequence of independent random variables with mean $\mu$ and finite variance . Define $S_n$ = $X_1 + X_2 +... +X_n$. (i) Show that law of large numbers doesn't hold for sequence ...
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1answer
27 views

Three pairwise uncorrelated random variables

Given $\xi$, $\eta$, $\zeta$ are pairwise uncorrelated, can we say, that $E(\xi\eta\zeta) = E\xi E\eta E\zeta$?
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65 views

Development of measure and probability theory

I am interested in a reference (article, maybe a book chapter) on the development of mathematical probability theory - that is, mostly starting from the beginning of the 20th century. It is surprising ...
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1answer
30 views

About random variable in Probability theory

I have a simple question. If $X_{1}$ and $X_{2}$ are two random variable on $(\Omega, P)$, they will be two functions, $f_{1}$ and $f_{2}$, from $\Omega$ to R. In case, $X_{1}$, $X_{2}$ are ...
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61 views

X is some random variable and f is a continuous function. Is f[E(X)] = E[f(X)]?

I am curious about at what conditions the expectation and a mapping could exchange their operation. Say, X is some random variable, and $f:R\rightarrow R$ is a continuous function. Does $$f[E(X)] = ...
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1answer
53 views

Is $e^{2(\cos(t)-1)}$ the characteristic function of some random variable?

I am asked to decide whether $$f(t)=e^{2(\cos(t) -1)}$$ is the characteristic function of some random variable. Attempt. I am trying to find directly a possible associated random variable (which ...
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1answer
42 views

Two definitions of the strong Markov property

In Durrett's textbook, the strong Markov property is defined as follows: For every bounded and measurable $\varphi$ and stopping time $N$: ...
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20 views

Splitting up Variance over components

If $f\colon [0,1]^n\to\mathbb{R}$ and we take variance wrt the Lebesgue measure(s), do we have $ Var(f)\leq\sum_{k=1}^n \int_{[0,1]^{n-1}}Var f^k_{\tilde{x}}\;d\tilde{x} $ where ...
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4answers
55 views

Tossing the coin $2n$ times, win if tossed heads $>n$ times

We have a coin with probability of heads $p = 0.48$, we toss it $2n$ times and win, if coin landed heads more than $n$ times. We can choose $n$. What $n$ should we pick?
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41 views

Set of product distribution is the cross product of its sections

Let $(X_1,X_2)$ be an independent random pair with distribution $F(X_1,X_2)$. Let \begin{align*} S&=\left\{F(X_1,X_2)\Big| F(X_1,X_2)=F(X_1)F(X_2) \text{ and } E[X_1^2] \le 1, \ E[X_2^2] \le 1 ...
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31 views

Identify the possible weak limit

Suppose $X_1, X_2, \ldots$ are independent random variables with distribution: $$ \mathbb{P}(X_n = 0) = \frac{1}{n}, \, \mathbb{P}(X_n = 2n) = 1 - \frac{1}{n} $$ Let $Y_n = \frac{X_1 + X_2 + \ldots + ...