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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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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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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18 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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77 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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36 views

What are examples of $f$ and $Y$, so that $\int_0^1f(t)Y_tdt$ is some known random variable

Is it possible to find an example of function $f$ and stochastic process $Y$, so that $\int_0^1f(t)Y_tdt$ is some known random variable (or random variable with known density, or characteristic ...
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1answer
15 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
40 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
29 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
26 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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16 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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199 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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30 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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33 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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28 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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33 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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43 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
14 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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24 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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105 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
31 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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16 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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16 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
59 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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30 views

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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58 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
21 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
20 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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45 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
27 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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3answers
58 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
42 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
37 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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7 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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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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1answer
35 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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26 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 + ...
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Is the set of product distributions with second moment constraint convex?

Consider the following set where $F(X_1,X_2)$ is distribution function of random pair $(X_1,X_2)$: \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, \ ...
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2answers
33 views

Limit of $P(X_n > a_n)$ where $X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$ and $a_n\xrightarrow[n \to \infty]{} \infty$

I've been working on following problem and could need some help. Let $X_n$ be a sequence of RV with $$X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$$ for some $\mu \in \mathbb{R}$ and ...
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1answer
47 views

Definition of the set of independent r.v. with second moment contstraint

I am trying to nice write the definition of the following set. Def: The set of all distributing of the pair $(X_1,X_2)$ such that $X_1$ and $X_2$ are independent Have second moment constraint ...
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14 views

Moment generating function vs Laplace transform

The moment generating function of a random variable $X$ is defined as $M_X(u)=E[e^{uX}]$ for $u\in\mathbb{R}$. On the other hand the (two sided)Laplace transform for the density $f_X(x)$ is ...
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1answer
20 views

Probability with qualifications and gender

Qualification Female Male Degree 5 1 None 5 4 School 8 12 Vocation 8 7 I've been going through some ...
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148 views

Probability of Distribution of Apples Question.

I have encountered this question which was actually assigned to a Biology class (the deadline has passed). It seemed simple at first but as more time passes by I realise how difficult it is. This is ...
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40 views

Find the conditional expectation $E[X|X\wedge t]$ [closed]

Let X be a random variable with exponential distribution with rate 1, $$f_X(x)=e^{-x}, x>0$$ Given $t>0$, denote $X\wedge t$= min{X,t}, find $E[X|X\wedge t]$ explicitly. Since $X\wedge t$ is ...
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49 views

Find an example of a sequence of random variables

Find an example of a sequence of integrable random variables $\lbrace X_n,n\geq1 \rbrace$ that has the following properties, $E[X_{n+1}\mid X_n]=X_n$ for every $n\geq 1$ but $E[X_{n+1}\mid ...
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22 views

ProbabilityTheory, precise definition of r.vs with same distribution

Studying probability theory, I wonder the definition of $X \overset{d}{=}Y$. However, it took me long time to search it to find nothing. Considering in the extension of limit law, i.e $X_n \to_d ...
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1answer
32 views

Student's distibution

If $X_i$ are independent equally distributed random variables, $S_n=X_1+...+X_n$, then $$\frac{S_n-n\mathbb E(X)}{\sqrt{n \sigma^2(X)}}$$ tends by distribution to $N(0,1)$ for $n \to \infty$. It is ...
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32 views

Let $E(X) = 1, E(X^2) = 3, E(XY) = -4$ and $E(Y) = 2$. Find $Cov(X, 2X + Y)$. [closed]

QUESTION: Let $E(X) = 1, E(X^2) = 3, E(XY) = -4$ and $E(Y) = 2$. Find $Cov(X, 2X + Y)$. I believe I have to use bilinearity to solve this, but I'm unsure as to how to go about doing that. Any help ...
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118 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
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23 views

When is the joint density differentiable

My question is the following: given a real random vector $X = (X_1,...,X_k)$ with differentiable marginal densities $f_1,...,f_k$, what extra conditions on the marginals are needed to ensure that the ...
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1answer
40 views

CLT for bounded and dependent sequence

Let $\displaystyle X_1,X_2,...X_n$ be identically distributed such that $\displaystyle Pr\{a \leq X_i\leq b\}=1$ for bounded constants $\displaystyle a,b$. Further Let $\displaystyle ...