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-...

learn more… | top users | synonyms (1)

0
votes
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) =...
4
votes
0answers
53 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
votes
1answer
37 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 \...
0
votes
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}...
3
votes
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$$m(\{x\}) = \alpha(x) - \alpha(x-)?$$
3
votes
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 ...
0
votes
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
votes
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 $1$?...
3
votes
0answers
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 ...
0
votes
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 ...
1
vote
0answers
67 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 ...
1
vote
2answers
35 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
votes
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}\...
0
votes
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 ...
1
vote
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
votes
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
votes
0answers
32 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\...
0
votes
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
votes
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. ...
1
vote
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 ...
1
vote
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 ...
1
vote
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
votes
1answer
22 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
votes
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 $\mu(A)$. ...
1
vote
0answers
33 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
67 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
0answers
27 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 ...
6
votes
1answer
71 views

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
8
votes
2answers
50 views

$\{f_n\}$ is uniformly integrable if and only if $\sup_n \int |f_n|\,d\mu < \infty$ and $\{f_n\}$ is uniformly absolutely continuous?

Let $(X, \mathcal{A}, \mu)$ be a measure space. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon$ there exists $M$ such that$$\int_{\{x : |f_n(x)| > M\}} |f_n(x)...
1
vote
3answers
37 views

How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ ...
2
votes
1answer
29 views

Bernoulli Trials: Law of Large Numbers vs Gambler's Fallacy, the N paradox

I have asked this question before but I think it wasn't clear what I implied with my succinct question, so I will be a bit more verbose this time. Lets set the following example: Bernoulli trials, K=...
0
votes
1answer
13 views

Use Bayes rule to test whether patient has disease after several positive tests

I have solved one of those standard bayes rule application exercises a la: Given a prevalence value of a disease, the sensitivity and the specificity of a test, calculate the probability that the ...
2
votes
1answer
25 views

Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...
0
votes
1answer
51 views

Infinite dimensional Borel-measurable function.

I am not quite sure how this statement about infinite dimensional borel-measurable functions is true, but the author says it is an easy observation: Let $D([0,\infty))$ denote the space of all ...
2
votes
0answers
41 views

How to physically model/construct a biased coin?

A perfectly unbiased coin is one that has the same probability for heads and tails (i.e., 50%/50%). A perfectly biased coin is one that has (as the name suggests) different probabilities for head ...
0
votes
0answers
23 views

When is a stochastic integral a martingale?

In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded ...
0
votes
2answers
30 views

Why is a continuous Lévy process twice integrable?

In his textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008), Jochen Wengenroth shows (p. 144) that if $(X_t)_{t\in[0,\infty)}$ is a continuous, real-valued Lévy process with $X_t\in \mathcal{L}_2$ ...
2
votes
0answers
18 views

How to compute integral of exponential martingale with respect to Poisson process?

Let $N=\{N_t:t\in\mathbb R_+\}$ be a homogeneous Poisson process with intensity $\alpha$ and $M_t=N_t-\alpha t$ the compensated process. I'd like to show that $N$ is not a natural process, i.e. that ...
-6
votes
1answer
69 views

Asymptotics of $\lim\inf X_n$, missing part of exam question [closed]

So I was working through an old exam and encountered the hilarious situation that part of the statement of the exercise was illegible. I was wondering if anyone could figure it out for me, so that I ...
3
votes
0answers
32 views

Are martingales progressively measurable? (Application to square integrable martingales)

This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively ...
4
votes
1answer
45 views

Bound for sum of normal distributions

I have encountered an exercise that was quite puzzling for me. Maybe someone can help me out here? So let $(X_n)_n $ be $N(-a,1)$ distributed, independent random variables where $a>0$. I need to ...
1
vote
0answers
51 views

Proof Attempt: Non-decreasing continuous CDF is standard uniformly distributed

Proof Attempt: For any random variable $X$ with non-decreasing continuous cdf $F(x)=\Pr(X≤x)$ (note that the inverse function does not necessarily exist due to flat regions in $F$), I wish to prove ...