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 [tag:probability-...

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1answer
22 views

Find variance of happy people sitting at $n$ regular polygon table

Let table has shape of $n$ regular polygon and at each side is sitting one person. Each person is flipping a fair coin once (results of $n$ independent tosses are independent). Person is happy iff he ...
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31 views

Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
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12 views

CGF determines the distribution

It is well known, that if the domain of the mgf of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments. However consider a Levy-...
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35 views

Probability question, can I reset the window or not

There is a wall street banker. The banker invests in a kind of share called as options. The main features of this share is as follows: You make a bet with a specified amount of information as to ...
2
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1answer
50 views

Find the limit of $P(\bar{X_n}\leq 1.8)$ for i.i.d random variables $X_i$s of known distribution

Let $X_1,X_2,…$ be a sequence of independent and identically distributed random variables with $P(X_1=1)=\frac{1}{4}$ and $P(X_1=2)=\frac{3}{4}$. If $\bar{X_n}=\frac{1}{n}\sum_{i=1}^{n}X_i$, for $n=...
3
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0answers
16 views

Information matrix for a Student's T distribution

I'm reading a paper from Creal, Koopman, Lucas "Univariate Generalized Autoregressive Score Volatility Models" and I'm stuck with this computation. $$ -\operatorname{E}_{t-1} \left[ \frac{\partial^...
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1answer
21 views

How to model guessing?

I want to model the knowledge of the student $i$, in a particular subject $S$. I give him a set of questions $Q$ from $S$ to test his knowledge. The level of his knowledge depends on the number of ...
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1answer
47 views

What Stochastic Calculi Other Than Ito And Stratonovich Exist?

When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale ...
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27 views

Does the pgf uniquely determine the distribution? [closed]

I know that the characteristic function of a random variable uniquely determines the distribution, but I'm just curious whether the probability generating function does so too(assuming that it exists)....
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2answers
22 views

Does this hold in every case, and if only this one, why? Expectation, mean of random variable.

Characteristic function of random variable $X$ let us denote as $f_X(t)$ and $EX$ it's mean or expectation. Does the following hold in all cases, because it keeps coming up and I don't know why it is ...
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1answer
26 views

Khinchin's Law of Large numbers proof unclarity.

This is the formulation: Let $X_n,n=1,2,...$ be independent, equally distributed random variables. $EX_k=a$(expectation) $k=1,2,...$. For this sequence of $X_n$ the law of large numbers applies: $$\...
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19 views

Find conjugate prior of an exponential family distribution

I read on Wikipedia that all exponential family distributions have conjugate priors. I have not, however, been able to find a reference that describes how to find it. So given $$f_X(x\mid\theta) = h(x)...
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4answers
92 views

Find the value of $\mathbb{E}(X_1+X_2+\ldots+X_N)$ of i.i.d random variables $X_i$s.

Let $ X_1,X_2,X_3 ,…$ be a sequence of i.i.d. random variables with mean $1$. If $N$ is a geometric random variable with the probability mass function $\mathbb{P}(N=k)=\dfrac{1}{2^k}$; $k=1,2,3,\...
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1answer
15 views

What is the chance of collision of latter group of card number among different people?

Let's assume we are a payment system issuing 16-digit cards. If we have X customers and issue Y cards, how to calculate the chance of at least single collision of last 4 digits within a single ...
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1answer
43 views

Define the function $g (y) = E[f(X,y)]$. Show that $g$ is Borel-measurable, and that $E[f (X,Y)|Y=y] = g(y)$

The original question is the number 10.6 of this pdf: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a bounded Borel-measurable function, and let $X$, and $Y$ be independent random variables. Define ...
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2answers
26 views

statistics- probability question [closed]

Let E be the event that a corn crop has an infestation of ear worms, and let B be the event that a corn crop has an infestation of corn borers. Suppose that P(E) = 0.24, P(B) = 0.16, and P(E and B) =...
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0answers
54 views

Inverse image is $\sigma$-algebra [on hold]

Let $(Y, \mathcal{A})$ be a measurable space and let $f$ map $X$ into $Y$, but do not assume that $f$ is one-to-one. Define $\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}$. How do I see that $\...
2
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1answer
38 views

Non-negative, integrable random variables which converge in probability and whose expected values have a finite limit

Suppose we have a sequence $X_1, X_2,...$ of non-negative, real random variables (not necessarily increasing) in $L^1$ which converge in probability to an integrable, non-negative random variable $X \...
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3answers
81 views

How can I compute $\mathbb{E}[Z^4]$ where $Z\sim N(0,1)$

Let $Z\sim N(0,1)$ and $Y=a+bZ+cZ^2$. I want to compute the variance of $Y$. This is what I did: $$\operatorname{Var}(Y)=0+b^2\operatorname{Var}(Z)+c^2\operatorname{Var}(Z^2)=b^2+c^2\operatorname{Var}...
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1answer
27 views

Lebesgue-Stieltjes measure corresponding to a right continuous increasing function, $m(\{x\}) = \alpha(x) - \alpha(x-)$ for each $x$

Let $m$ be Lebesgue-Stieltjes measure corresponding to a right continuous increasing function $\alpha$. How do I see that for each $x$, we have$$m(\{x\}) = \alpha(x) - \alpha(x-)?$$
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0answers
41 views

Knight (Chess) Problem on telephone keyboard

There is phone keyboard with Knight on 0 (as shown below). 123 456 789 0 Knight moves as per the rules of chess (2 straight and one turn). T is no. of moves ...
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1answer
38 views

What does a random variable 1 with subscript [0,1/2] mean?

I came across the following notation that I cannot follow: $1_{[0,1/2]}$ It is supposed to be some kind of random variable (or just an event? not sure) It is hard to google this, too. What does such ...
3
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1answer
65 views

The Uncountable and Probability

Suppose we draw a random uniformly number from $[0,1]$, if we do this countable many times, how many times will we get $1$, I suspect $0$? If we do it uncountable many times, how often will we get $1$?...
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34 views

Showing that $\sigma$-algebra is uncountable [duplicate]

Suppose $\mathcal{A}$ is a $\sigma$-algebra with the property that whenever $A \in \mathcal{A}$ is nonempty, there exist $B$, $C \in \mathcal{A}$ with $B \cap C = \emptyset$, $B \cup C = A$, and ...
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2answers
21 views

Probability of two statistically independent, uniformly distributed variables occurring within time frame of each other?

Say two events will occur independently of each other, only once each. The time of each event occurring is uniformly distributed from 0 to 10 seconds. What is the probability that the events will ...
2
votes
0answers
25 views

Distribution of marginal Wiener process.

Let $(W^1,W^2)$ be a two-dimensional Wiener process with correlation $\rho$. Let $\mathcal{F}^1_t$ be the filtration generated by $W^1$ up to $t$. I would intuitively think that for $h$ measurable ...
4
votes
1answer
32 views

Indicator function for limsup, liminf [duplicate]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
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0answers
76 views

For Which General Distributions Does This Inequality Hold?

Let $X$ be a random variable with mean $\mu$, where $0 < \mu < 1$. Let $X(n)$ be the sum of $n$ independent ,identically distributed, $X$ variables. Under what conditions on $X$ , possibly ...
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2answers
36 views

Is there a book or lecture notes on Percolation Theory containing exercises?

I have seen Grimmett's Percolation Theory and I have also seen a few online lecture notes. But they don't have exercises. I understand it is stupid to ask of exercises in such a recent and hot ...
4
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1answer
1k views

Inverse Mills ratio for non normal distributions.

We have the well known result of the inverse Mills ratio: $$ \mathbb{E}[\,X\,|_{\ X > k} \,] = \mu + \sigma \frac {\phi\big(\tfrac{k-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{k-\mu}{\sigma}\...
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0answers
35 views

$(B_t)_{t\ge 0}$ be Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t}$ is an analytic function. [on hold]

Let $(B_t)_{t\ge 0}$ be a one-dimensional Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t} \; \text{for all} \; t\ge 0, \xi \in \mathbb{R}$ is an analytic function. A more general question ...
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0answers
22 views

Irwin - Hall distribution of n different uniform distributions $U_k(a_k,b_k)$

Given $n$ independent and identically distributed uniform distributions $U(0,1)$, their sum is: $$X=\sum_{k=1}^nU_k(0,1)$$ The $pdf$ of this sum is given by the well known Irwin - Hall distribution. ...
2
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1answer
42 views

Why Characteristic function is given such a name?

I've come to know that, For random variable $X$ ,with a Probability mass function P, $\phi_X(t)$ defined by : $\phi_X(t) : \mathbb R \to \mathbb C$ $\phi_X(t) = E(e^{itX}) =E[\cos tX + i\sin tX]$ ...
4
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0answers
33 views

Asymptotic distribution of $(\bar{Y}_n - E(|X_1|)) / \bar{X}_n$

Let $(X_n)$ be i.i.d double exponential random variables, with PDF $\frac1{2a}e^{-|x|/a}$, thus $E(X_n) = 0$, $E(|X_n|) = a$ and $V(X_n)=2a^2$. Consider the sample means $$\bar{X}_n=\frac1n\sum\...
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1answer
39 views

distributing numbered balls with duplicates into 4 boxes [closed]

How many ways are there to distribute 52 balls, numbered 1 to 13 with 4 duplicates for each number, into 4 distinguishable boxes.
3
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3answers
29 views

Why does $A_1^\text{c}$ have an infinite number of measurable subsets?

Let $\mathcal{A}$ be a $\sigma$-algebra. Show that if $|\mathcal{A}| = \infty$, then $\mathcal{A}$ is uncountable. We want to construct an infinite sequence of nonempty disjoint measurable sets. ...
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2answers
14 views

Dependency of function of independent random variables

$X$ and $Y$ are independent and identically distributed random variables, $c$ is a constant. I wonder if $\frac{1}{X+c}$, and $\frac{1}{Y+c}$ are independent? In other words, are the functions of ...
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0answers
18 views

Equality of random variables measurable w.r.t. different sigma-algebras

I'm stuck trying to prove the following statement: Let $\tau $ be a non-negative random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. We'll consider the following ...
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0answers
12 views

powers of the transition matrix for a subshift of finite type

Is there a relation between powers of the transition matrix for a subshift of finite type and iterations of the (forward) subshift? It is very nice to understand the transition matrix via a graph and ...
2
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1answer
23 views

Square Integrable Martingales and the Unit Process

Let $X_t$ be a continuous square-integrable martingale. Then it is true (I think, please correct me if I am wrong) that: $$\forall t \in [0,\infty), \quad \int_0^t 1_{[0,t]}(s) dX_s = X_t - X_0$$ So ...
3
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1answer
32 views

Does it follow that $\mu$ is a measure? [duplicate]

Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Cal the limit $\mu(A)$. ...
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0answers
34 views

Proving sigma-algebras equality

I'm not sure if my proof of the following statement is correct. Let $\tau : \Omega \to \mathbb{R}_+$ be a non-negative random variable, defined on a probability space $(\Omega, \mathcal{G}, \...
2
votes
1answer
69 views

Sequence of Erdos-Renyi random graphs convergent with probability 1

Definitions Let $H,G$ be finite simple graphs. Then the density of $H$ in $G$, denoted $d(H,G)$, is defined as the probability that a randomly chosen $|H|$-tuple of vertices of $G$ induce a graph ...
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0answers
12 views

How does commutative property play a big role in probability theory? Kindly, explain why do we need free-independence?

I got to know that as matrix operations are non-commutative, calculation of moments are not possible? To be able to do so, one need to ralax the independence to something called free-independence. I ...
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3answers
75 views

probability of having an ace on each hand when dealing 52 cards to 4 hands

The question from DeGroot's book: "probability and statistics" "Suppose that a deck of 52 cards containing four aces is shuffled thoroughly and the cards are then distributed among four players so ...
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0answers
14 views

What is Steiltjes transform (other Integral transform) and how does it helps in probability theory, specifically in random matrix theory?

I have started growing interest in random matrix theory. Trying to understand it from "Random Matrices" by Madan Lal Mehta and "An Introduction to Random Matrices" by Anderson and many sources on ...
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1answer
41 views

Help understanding convolutions for probability?

I have been trying to do some problems in probability that use convolutions but there has not been much of an explanation of what a convolution is or the purpose of using a convolution. For example ...
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0answers
34 views

Simplifying Multiple Integral for Compound Probability Density Function

Are there any ways to simplify this multiple integral? $$ \hat{f}\left(\left.y\right|\alpha\right)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\hat{f}\left(\left.y\right|\theta_{1}\right)\hat{...
2
votes
1answer
57 views

How to prove that if $E[X^2]$ is finite then $n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0$?

Let $X$ be a random variable with $E[X^2]<\infty$. I want to prove that $$ n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0 \text. $$ I tried to apply Chebyshev's inequality, ...
0
votes
0answers
33 views

Given $n$ heads out of $n$ tosses. What is the posterior probability that coin is fair? [closed]

I am given an $\sigma$-fair coin with the probability of head $(\theta)$ being in the interval $[\frac{1}{2} - \sigma, \frac{1}{2} + \sigma]$. Also I am given: For a Bayesian analysis of the ...