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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An example of stochastic process

I use the following definition for a stochastic process. Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection ...
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32 views

$F_n \overset{w}{\to} F$, and $F$ is continous. Show that $F_n$ converges to $F$ uniformly on $\mathbb{R}$

$\{F_n\}$ and $F$ are distribution functions, and $F$ is continuous on $\mathbb{R}$. If $F_n$ converges weakly to $F$, show that $$ \sup_x | F_n(x) - F(x) | \to 0, n \to \infty $$ I know that ...
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Probability that a word contains at least 3 same consecutive letters?

Assume we have a word of length $n$ and an alphabet of length $26$ (the small letters a through z, if you want so. How likely is it that this word contains at least $k := 3$ consecutive letters of ...
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57 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
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71 views

Expected maximum score(dice).

Given a dice with m faces, such that face_i has i dots, and each face appears with probability 1/m. What is the expected maximum number of dots she could get after tossing the dice n times.
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Given “N” number of events, calculate the number of condition to check whether all the events are statistically independent.

please help me out here, i dont even know where to start with this question :(. Any guidelines anything at all that may give me an idea to answering the question will be greatly appreciated. Please ...
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53 views

Prove that a.s.$\lim\limits_{t\to\infty}\frac{N_t}{t}=\frac{1}{\mu}$

Consider a diligent janitor who replaces a light bulb the instant it burns out. Suppose that the first bulb is put in at time zero and let $X_i$ be the lifetime of the i-th bulb. Suppose ...
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30 views

Prove that the discrete time martingale can be represented by $E (Y_{n +1} \mid F_n) = 0$ if $Y_{n +1} = X_{n +1}-X_n$, for $n = 0,1, \ldots $

Prove that the discrete time martingale can be represented by $E (Y_{n +1} \mid F_n) = 0$ if $Y_{n +1} = X_{n +1}-X_n$, for $n = 0,1, \ldots $ I want to use the sequence $(y_n)$ called "martingale ...
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43 views

Calculate the Probability in competition

The committee RAM competition knows from experience that the probability of successfully Contest is 0.95 for the student who has grade "very good" in BAC test , 0.5 one who has 'Good' in ...
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37 views

Joint distribution of random variables

Let $S$ and $T$ two independent random variables. Suppose that S is a standard Gaussian random variable with density $f(s)=(2\pi)^{-1/2}e^{-s^2/2}$ and T with, $2f(t)\mathbf 1_{\{\mathbb ...
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$P_{x}(T_{B_{0,r}}<\infty)$ in integral form [closed]

$$P_{x}(T_{B_{0,r}}<\infty)\tag1$$ for $x\in (B_{0,r})^{c}$ in three dimensions for Brownian motion $\textbf Q_{1} $ Is there a way to get (1) in an integral form or at least relate it to one? In ...
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40 views

Probability of getting head in coin flip [duplicate]

Suppose a football match is going to be started. The referee should flip a coin to give the ball to one of the football teams. In front of the two captains the referee flips the coin two times and ...
2
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1answer
47 views

random variables not independent but $\mathrm{E}[X|Y]=\mathrm{E}[X]$

I have to find two r.v. X,Y defined in a probability space ($\Omega, \mathcal{F}, \mathrm{P}$), which are not independent but for which $\mathrm{E}[X|Y]=\mathrm{E}[X]$ nonetheless, with ...
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1answer
40 views

Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ converges to 1 in some neighbourhood of $t=0$.

$X_n$ is a sequence of random variables, and $\{\varphi_n(t)\}$ is the corresponding sequence of characteristic functions. Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ ...
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24 views

Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
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2answers
43 views

How to show that measurable functions preserve mean independence?

Let $X$ and $Y$ be continuous random variables such that $$\mathbb{E}(X\mid Y=y)=\mathbb{E}(X) $$ for all $y\in \operatorname{Supp}(Y)$. How can I show that for any measurable function $f$ we have ...
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1answer
32 views

Random variable bounded by another random variable

How to find $\Pr(z<X<Y)$ if $X$ and $Y$ are independent exponential r.v.'s with parameters $\lambda$ and $\mu$ So $x$ is bounded by $z$ and $y$, and y must be go from $z$, (and not from ...
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0answers
33 views

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$. [duplicate]

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$. From here http://math.stackexchange.com/a/878635/140308 (proof attempt is there too) Sorry ...
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exponential bound on the number of possible clusters at $0$ in $\mathbb{Z}^d$

Let us say that $\mathbb{Z}^d$ is given the usual lattice structure as a graph. I want to know the number of connected induced subgraphs of size $k>0$ that include the vertex $0$. Call this ...
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22 views

Green's function of a Markov Chain, and maybe of a Feller Process?

How are the Green's functions of a Markov chain related to the notion from PDE theory? For instance, if the Markov chain (i.e. discrete state space) is continuous time, then the Green's function I'm ...
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1answer
47 views

Proving independence of random variables

If $X$ and $Y$ are independent exponential random variables with parameter $\lambda$ and $\mu$. Let $Z=\min(X,Y)$, prove that $Z$ and $\mathbf 1_{\{X<Y\}}$ are independent. I don't know, how ...
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23 views

How to find the dynamics of stochastic process?

We have $Y_t=e^{\int_0^t W_sds}$. How do I obtain the dynamics of $Y_t$ (i.e. $dY_t$)? It seems that we can't use Ito Lemma because $\int_0^t W_sds$ is not in the form $X_t = \int_0 ^t \sigma_s dW_s ...
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25 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
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1answer
33 views

How does arg max work in this context?

I'm implementing some stuff for machine learning and I ran across this post detailing some information on Bayes Theorem I was looking for: ...
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1answer
30 views

Dynamics of short rate in HJM

According to a simplified HJM framework, we have: Forward Rate: $f(t,T)=\sigma W_t +f(0,T) +\int_0^t{\alpha(s,T)}ds$, where $W_t$ is brownian motion. Dynamics of forward rate: ...
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31 views

exercise 1.21 of chapter 1 of Revuz and Yor's

This is the exercise 1.21 of chapter 1 of Revuz and Yor's: Let $X=B^+$ or $|B|$ where $B$ is the standard linear BM, $p$ be a real number $>1$ and $q$ its conjugate number ($q^{-1}+p^{-1}=1$). ...
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35 views

Mean time spent in a state in a CT Markov Chain

I consider a continuous-time homogenous Markov chain: with discrete state $X$ taking values in $\mathcal{F}=\{1,\cdots,N\}$ with the transition rates satisfying: \begin{equation} \begin{cases} ...
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66 views

Probability two people born on 1 April

Find the smallest number of people you need to choose at random so that the probability that at least two of them were both born on April 1 exceeds 1/2. My answer: Total of n people chosen at random ...
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1answer
17 views

Convergence in distribution for changing domains.

I am trying to consider whether this is possible and/or reasonable: Let $X_n:\Delta_n \to \mathbb{R}$ be a sequence of random variables, defined over a unique space $\Delta_n \subseteq \Omega$ for ...
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1answer
32 views

Sum of Independent RVs in $L^1$

My friend and I were studying for a preliminary exam in probability and came across the following problem. Suppose $X$ and $Y$ are independent random variables. Prove that if $X+Y\in L^1$ then $X$ ...
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1answer
28 views

Probability density function of two uniformly distributed stochastic variables

I'm currently stuck on an exercise involving two independent stochastic variables X and Y. Both X and Y ~ U(0,1) (uniform distribution) The goal of the exercise is to calculate the probability ...
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1answer
30 views

$\sigma(X_1,\dots, X_n) = \sigma(S_1,\dots, S_n)$ generated by partial sums

is true that $\sigma(X_1,\dots, X_n) = \sigma(S_1,\dots,S_n)$ where $S_n=\sum_{i=1}^n X_i$ in general or I have to impose additional restrictions to the random variables (for instance, independence)? ...
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26 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
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55 views

Are the product and ratio of two characteristic functions still characteristic functions?

Let $\bf X $ and $\bf Y$ be random vectors (that may be dependent). Let $\varphi_{\bf X}(\bf t)=E[e^{i\sum X_it_i}]$ be the characteristic function of the random vector $\bf X$. (1) Is $\varphi_{\bf ...
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33 views

Is $E[Bin(X,p)]=E[X]p$?

We have given some random variable $X$ with mean $E[X]=:\mu$. Now we are interested in a random variable $Y \sim Bin(X,p)$. Is it true that $$E[Y]=\mu p?$$ What confuses me is that normally the ...
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1answer
40 views

Inequality on characteristic functions (probability theory)

Show that for every real characteristic function $\phi(t)$ we have $$1-\phi(2t) \le 4(1-\phi(t))$$ I am not sure where to begin. It seems I am missing some formula or theorem, or is it really that ...
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92 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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15 views

L1 expected error for scale/translation in densities

This is an example given in the article about testability (Devroye and Lugosi 1999) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
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1answer
21 views

Comparison between these Ito Lemma versions

According to wikipedia : I found another version : Please explain the difference for me.
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87 views

Expected number of coin tosses until a run of $k$ successive heads occurs

Suppose each coin toss is independent, what is the expected number of coin tosses until a run of "k" successive heads occur? Tried finding a recursive expression to solve the problem but got ...
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Birthday “Paradox” - another, different, version!

Background Many people are familiar with the so-called Birthday "Paradox" that, in a room of $23$ people, there is a better than $50/50$ chance that two of them will share the same birthday. In its ...
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Cauchy sequence of random variable

I know (from calculus) the Cauchy convergence theorem for sequence of real numbers...how to show it with random variables? I have $S_n=\sum_{i=1}^n X_i;$ $X_i$ being iid centered r.v., $S_n$ is a ...
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Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
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Help proving $Pr(\mathcal{X})= \phi_1(X,Z)\phi_2(Y,Z)$ if $ P \models (X \perp Y | Z)$ and $\mathcal{X}=X \cup Y \cup Z$

I was trying to prove the following: if $X,Y,Z$ were three disjoint subsets of variables such that $\mathcal{X}=X \cup Y \cup Z$, Prove that $ P \models (X \perp Y \mid Z)$ if and only if we can ...
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2answers
31 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
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1answer
176 views

Mathematician who talked about the probability of a “good” graph?

In my undergraduate years, one of my professors always talked about this one mathematician who was talking about "good" graphs and wondered about the existence of such a graph. Apparently this ...
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Is there a set of 69 length-6-sets out of 46 numbers [1..46] so that those length-6-sets “cover” all possible 1035 length-2-sets of 46 numbers?

1.) For this question, we have 46 numbers (balls, cards, whatever): {1,2,3,4 .... 45,46} ======================= 2.) Each length-6-set of 46 numbers ( e.g. {1,2,3,4,5,6} or {1,13,16,17,32,46 } ...
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21 views

Integration respect the Lévy measure

How I can compute numerically $$\int_{a}^{b}f(z)\nu(dz)$$ where $\nu$ is a Lévy measure and $f$ is a continuous function? Is it equivalent to $$\int_{a}^{b}f(z)d\nu(z)$$ Thanks
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79 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
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10 views

Generate quadrature points from a distribution

Is there any method to generate quadrature points from any arbitrary probability distribution, $p_{X}\left(x\right)$? We already know about Gauss Hermite rule for Normal distribution, Gauss-Laguerre ...