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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Exclusion-Inclusion principle for Hitting times of two disjoint sets

Consider disjoint sets A, B and Brownian motion $B_{t}$ with $B_{0}\notin A\cup B$. Let $T_{A}:=inf_{t>0}\{B_{t}\in A\}$. Then, do we get $P(T_{A\cup B}<\infty)=P(T_{A}<\infty)+P(T_{ ...
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36 views

Minimum of a random variable sequence

$S_{n}$ model the price of a financial asset. The recurrence relation is given by: $$ S_{n+1} = (1 + r\Delta t_{n} + \Delta W_{n})S_{n}, n = 0, \dots, N $$ where $\Delta W$ has a normal ...
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True or false: If a distribution has a conjugate prior, then it is a member of the exponential family.

I would like to know if it's true that "A distribution has a conjugate prior if and only if it is a member of the exponential family". I know that all members of the exponential family have conjugate ...
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55 views

Prove $X_n = \frac{1}{a_n} 1_{(0,a_n]}$ a martingale

With the following premises, I want to prove, that the series of random variables $(X_n)_{n \in\mathbb{N}}$ is a Martingale: Let $\Omega := (0, 1] \subset \mathbb{R}, \mathfrak{F}$ the ...
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35 views

Extension of measures

Here an exercise of the book: A probability path by Sidney Resnick. Suppose $P$ is a probability in a $\sigma$-field $\cal{B}$ and $A\notin \cal{B}$. Let $\mathcal{B}_1:=\sigma(\mathcal{B}\cup\{A\})$ ...
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Joint distribution expressed with conditional distribution.

Let $(\Omega,\mathfrak{A},\mathbb{P})$ be a probability space, $(\Omega',\mathfrak{A}')$, $(\Omega'',\mathfrak{A}'')$ two measure spaces and $$X\colon(\Omega,\mathfrak{A},\mathbb{P})\rightarrow ...
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53 views

Prove that $P(A \cap B) \leq P(A) + P(B)$?

I need to use theorem 2.7 which says that : $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. I started by solving the above equation for $P(A \cap B)$. I got: $P(A \cap B) = P(A) + P(B) - P(A \cup B)$. ...
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70 views

More general definition of expected value

Let $X$ be a random variable with pdf $f$. I would like to know why: $$\operatorname{E} [X] = \int_\Omega X \, \mathrm{d}P = \int_\Omega X(\omega) P(\mathrm{d}\omega)= \int_{-\infty}^\infty x f(x)\, ...
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36 views

Some hints on how to prove these two questions?

I was told to draw out a picture (Venn Diagram) and then to use them to prove that $P(A) \geq P(A \cap B)$ $P(A) \leq P(A \cup B)$ I have no idea where to begin and the pictures aren't helping ...
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1answer
23 views

A LLN type theorem on the supremum of functions of a RV

Let $X_1,\dots,X_n$ be iid real valued random variables. Let $\mathcal{F}$ be a set of functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\mathbb{E}f(X_i) < \infty$ for all $f \in ...
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14 views

Scheduling Algorithm for a multi-server queue problem

I have 4 servers, n customers and m reports. At any time, a customer may request one of m reports. There are only 4 servers which are capable of generating reports. Each server can only process one ...
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1answer
44 views

A basic probability problem

Is it possible to give reference (like is there any name for it, where it is used, some simplification for its expression in terms of $E[X]$) for the quantity $\ln E[e^X]$. I know that it is $\geq ...
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2answers
57 views

How to show these two random variables are not equal with positive probability?

Given iid $X_1, \dots, X_n \sim N(a, a^2), a \neq 0$, how can we show that it is not true $\frac{(\sum_i X_i)^2}{n(n+1)} = \frac{\sum_i X_i^2}{2n}$ a.e.? My thought was checking the moments. I got ...
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46 views

One Question on Law of Total Probability

Let $(X_n)$ with $n \in \mathbb N_0$ be a discrete martingale. Then I read the following identity which is said to be derived from the law of total probability. $$ \mathbb EX_m = \left( \sum_{n=0}^m ...
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28 views

Prove: Quotient of two N(0,1) is Cauchy(0,1)

Problem: Show that if X and Y are independent N(0,1)-distributed random variables, then X/Y ∈ C(0,1). Question: I don't know how to proceed below. I want to prove that the PDF of X/Y is Cauchy. PS. I ...
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14 views

Understanding of conditional density functions, continuous case

I'm not really measure theory-saavy, but I'm trying to understand, at least in an informal, intuitive level, what exactly allows us to define a conditional density in the continuous case. It would be ...
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45 views

Proof/derivation of the sum of a sequence

Question along the lines, "Use a sum formula to derive a closed form formula" for a section covering combination theorems $$ \mbox{Integer function:}\quad {\mathrm f}\left(\,n\,\right) =1 + 4 + 7 + ...
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60 views

If $Y\ge 0$ almost surely and $X+Y \sim X$ then $Y=0$ almost surely

Let $X, Y$ be random variables on the same probability space such that $Y \ge 0$ almost surely and $X+Y$ and $X$ have the same distribution. Please resolve whether these conditions imply that ...
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1answer
32 views

Conditional expectation of symmetric Sigma algebra

Another exercise with conditional expectation that I have problems with. Let $\Omega=[-1,1]$, $\mathcal{F}=\mathcal{B}(\Omega)$, $\mathbb{P}=\frac{1}{2}\lambda$. Let X be a ...
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38 views

Finding all Borel measures $\mu_X$ such that $Y\sim \mathcal{N}(0,1) \Rightarrow XY \sim \mathcal{N}(0,1)$.

Find all Borel measures $\mu$ on $\mathbb{R}$ such that for every independent random variables such that $X \sim \mu$ and $Y\sim \mathcal{N}(0,1)$ we have $XY \sim \mathcal{N}(0,1)$. To be honest ...
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38 views

CDF calculation

Help me please with the following question. I'm reading Gill's lectures on survival analysis http://www.math.leidenuniv.nl/~gill/stflour0.pdf On the page 27 he states that: by the Glivenko-Cantelli ...
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41 views

conditional probability exponential distributed

Let $X_1,\ldots,X_n$ be independent random variables, $X_i \sim{}$ $\mathrm{exponential}(\lambda_i)$. Let $X=\min\limits_{1\le i \le n} X_i$. Calculate $\mathbb{P}(X=X_i)$ At first I ...
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29 views

How do you transform probabilities from the form P[X=x] to P[X =< x]

I'm working on a problem that requires you to use a binomial distribution to solve the problem. Now we want to determine x such that P[X > x] =< 0.01 or, ...
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32 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
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1answer
14 views

Consistency of kernel density estimator with constant bandwidth

Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is \begin{align} \hat{f}_h(x) = ...
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13 views

Asymptotic uniform integrability and moments of Student's $t$

I am working on an exercise where I am trying to show that the moments of a $t$ distribution converge in probability to the moments of a standard normal. I'm in need of help with what I think is the ...
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1answer
27 views

If the random variable $X$ is standard Cauchy then so is $1/X$

Problem Prove that $X \in C(0,1) \Rightarrow 1/X \in C(0,1)$ where $C$ is the cauchy distribution. Attempt I try to prove they have the same density function. Question Is my proof correct?
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37 views

Almost sure limit of $\log(X_1 + X_2 + … + X_n) - \log(n)$

Let $X_n$ be an i.i.d. sequence of positive random variables with expectation 2 and variance 1. What is the almost sure limit of $$\log(X_1 + X_2 + ... + X_n) - \log(n)$$ as $n \to \infty$ Would it ...
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26 views

Show that the collection of all subsets of a finite sample space satisfies the 3 conditions to be called a collection of events.

As title. Suppose that the sample space $S$ of some experiment is finite. Show that the collection of all subsets of $S$ satisfies the three conditions required to be called the collection of events. ...
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64 views

Mutually exclusive dices?

Suppose I have 3 dice. Each has some mechanism that can prevent other dice being in 2 if itself is 2 when they are rolled together. Now I roll the 3 dice at the same time. Then what is the probability ...
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59 views

Probability generating function of the sum of two random variables

Let two $\mathbf{N}$-valued random variables $X$ and $Y$ be given, and let $\phi_X(s) = \sum_k \mathbf{P}(X = k) s^k$ and $\phi_Y(s) = \sum_k \mathbf{P}(Y = k) s^k$ be their respective probability ...
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25 views

For a set of size n, how many collections from the set include a particular element?

I'm stuck on a problem which can be simplified as below: Given the first 12 letters of the alphabet (A .. L), in how many collections of 4 letters do both 'A' and 'B' appear. I recognise that there ...
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1answer
34 views

What is the Conditional law?

I am sorry to ask such a question, but I always cant understand what is meant by the notation I am currently reading. In a mathematical paper I witness a formula as such $\mathcal{L}\left( ...
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1answer
31 views

Equivalence in Portmanteau's lemma

I'm trying to understand the proof of one of the equivalences in Portmanteau's lemma. The equivalence (or implication rather, as that's what I'm trying to prove) is this: For any random vectors ...
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57 views

How to show that $\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$

Let $x_n$ be a sequence of reals. Show that $$\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$$ Since the weak convergence is equivalent to pointwise convergence of characteristic functions ...
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Ito formula for jump proccess

I have just learned Ito fomula for jump processes but I have still not understood it well. Assume that I have $dS_r=S_{t^-}\mu+S_{t^-}\sigma dB_t +S_{t^-}\int_{\mathbb{R}^+}(y-1)N(dt,dy), \;\; 0\leq ...
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51 views

A random number of random variables, (expectation help)

everyone First a definition let $S_N = X_1 + X_2 \cdots + X_N$ where $X_i$'s are random variables and $N$ is also a random variable. Also assume that the $X_i$'s(integer valued),independent ...
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probability of an event with countably many sample points

The following is Problem 1.9 from A Modern Approach to Probability Theory. Let $A$ be an event which contains countably many sample points. Assume that each of the sample points in $A$ is an event. ...
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81 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
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27 views

Convergence of expected values as random variables converge almost surely

Let I have a sequence of random variables $X_n$ that converges to random variable $X$ almost surely as $n\to\infty$. How can I proof that $\lim_{n\to\infty}\mathcal{E}[X_n]=\mathcal{E}[X]$ where ...
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Expected values of continuous and bounded functions are equal then random variables are equal, too.

I have seen several of reasoning based on the following fact: Real random variables $X, Y$ in $\mathbb{R}^n$ are equal almost surely if and only if $\mathbb{E}g(X)f(X) = \mathbb{E} g(X)f(Y)$ for ...
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38 views

Conditional Expectation and Conditional Independence

Suppose we have 3 $\sigma$ algebras A, B and C such that A is independent of C. The random variable X is measurable with respect to $\sigma$(B,C), the $\sigma$ algebra generated by B and C. Is it true ...
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32 views

Can this function be a density function of a continuous random variable X?

F(x) = 0, if x < 1 F(x) = 1, if 1<=x<=2 F(x) = 0, if x>2 I think it could be, as long as the integral is 1. Any ideas?
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65 views

Solve $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t) \, dt + \sqrt{1+X_t^2} \, dW_t$ explicitly

Solve explicitly the 1-dimensional equation: $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t)dt + \sqrt{1+X_t^2}dW_t$ I have hopelessly been guessing solutions to this. Does anyone know how to solve this ...
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1answer
36 views

Markov processes on function spaces

Is there any reference on Continuous time Markov process whose state space is infinite dimensional function spaces, such as the space of continuous functions $C(R^d)$? It seems Dirichlet Form is a ...
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22 views

Problem EA 13.2 from David Williams' Probability with Martingales

I am stuck trying to solve this problem from Williams' Probability with Martingales: My attempt: $E(X_n) = E(e^{aS_n - bn})$ $= e^{-bn}E(e^{aS_n})$ (because $e^{-bn}$ is not random) $= ...
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1answer
51 views

Markov Chains : Can anything be said about what happens in between two transition?

In time homogeneous discrete Markov chains we take a set period for a single transition. In examples we see sometimes depending on the examples the transition period being a a month a week etc. I'm ...
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1answer
14 views

How to calculate the median and the quantiles of this distribution?

A median of a distribution of one random variable $X$ of the discrete or continuous type is a value of $x$ such that $P(X < x) \leq 1/2$ and $P(X \leq x) \geq 1/2$. If there is only one such $x$, ...
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27 views

Sample Mean and Sample Variance

Consider a sample of data S obtained by flipping a coin x, where 0 denotes the coin turned up heads, and 1 denotes that it turned up tails. S = {1, 1, 0, 1, 0} What is the sample mean for this data ? ...
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Geometric and binomial distribution problem

Let $X \sim Bi(n,p)$, and $Y \sim \mathcal{G}(p)$. (a) Show that $P(X=0)=P(Y>n)$. (b) Find the number of kids a marriage should have so as the probability of having at least one boy is $\geq ...