0
votes
0answers
15 views

Cesaro means converges in first mean to 0.

I would appreciate any suggestions to prove the following statement If $\{ X_i: i=1,..\}$ is a sequence of independent uniformly integrable (U.I) random variables ...
1
vote
0answers
25 views

Soft Question: What are some elementary motivations of using functional analysis to study probability theory?

Recently i've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure theoretic ...
1
vote
1answer
41 views

proof about sigma-algebra

The question is that, $f$ is a function mapping $\Omega$ to another space $E$ with a $\sigma$-algebra $\varepsilon$. Let $\mathbf{A}= \{A\subset\Omega : \text{there exists } B \in \varepsilon \text{ ...
0
votes
0answers
30 views

Sum of function applied to parts not equal to function of total

The general goal is to determine the effectiveness of the test pill's ability to keep the test subjects from getting sick using the following data. | Test Subjects | Took Test Pill | ...
1
vote
0answers
26 views

Show that $ \prod_{k=1}^n [1 + p_{nk}(e^{it} - 1)] \rightarrow e^{\lambda(e^{it} - 1)}, n \rightarrow \infty $

Suppose that $0\leq p_{nk} \leq 1, 1 \leq k \leq n$, $\max_{1 \leq k \leq n} p_{nk} \rightarrow 0, n \rightarrow \infty$ and $\sum_{k=1}^n p_{nk} \rightarrow \lambda$. Show that $$ \prod_{k=1}^n ...
0
votes
2answers
33 views

Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
0
votes
1answer
24 views

Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
6
votes
1answer
201 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
0
votes
1answer
21 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
1
vote
0answers
39 views

Central limit theorem does not converge to random variable

Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ...
0
votes
0answers
11 views

Is it possible to get the PDFs of each of the three vector components knowing the PDF of the modulus if isotropy is guaranteed?

The PDF of a given vectorial quantity modulus is known. I would like to obtain the PDF of each of the three vector components in the case of isotropy, i.e. the three PDFs are supposed to be equal and ...
1
vote
1answer
56 views

Eigenvalue markov chain

I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ...
2
votes
1answer
41 views

a question about Kolmogorov's Existence Theorem

I (a beginner in probability) have some confusions arising from problem 36.7 of the book "probability and measure" by Billingsley. It says that there is on the unit interval with Lebesgue measure no ...
1
vote
0answers
42 views

Transient/Recurrent Markov chain

I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ...
0
votes
1answer
30 views

Is this a Markov chain property

For $A,B$ measurable sets and $(X_n)_n$ a Markov chain. Do any of the following properties hold? $$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$ or $$P(X_2 \in B|X_1 \in A,X_0=x_0) = ...
0
votes
0answers
33 views

Strong Markov property and its meaning

Given a sequence of random variables $(X_n)_n$ (fulfilling the Markov property) and a stopping time $\tau$ such that $P(\tau < \infty)=1$, we have that ...
0
votes
0answers
26 views

The maximum term in Binomial distribution [duplicate]

In Binomial distribution, if $p = q = \frac{1}{2}$. The maximum of $C_{2n}^k p^kq^{2n-k}$ is easy to find,It is equal to $C_{2n}^n2^{-2n}$. and use Stirling's formula $n!\sim\sqrt{2\pi n}\exp(-n)n^n$. ...
0
votes
1answer
17 views

Asymptotic in hypergeometric distribution.

Assume $n_1 + n_2 = n$ and $M_1 + M_2 = M$. then $$\frac{C_{M_1}^{n_1}C_{M_2}^{n_2}}{C_{M}^{n}} \rightarrow \frac{n!}{n_1! n_2!}p^{n_1}(1-p)^{n_2}$$ when $M\rightarrow \infty$ and $M_1 \rightarrow ...
1
vote
3answers
35 views

Methods to distinguish continuous probability distributions

I read in the Wikipedia article for Variance The variance is one of several descriptors of a probability distribution. In particular, the variance is one of the moments of a distribution. In that ...
0
votes
1answer
40 views

Fubini Question in context of Independence

I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ...
1
vote
1answer
32 views

American Put question

If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option.
0
votes
1answer
65 views

conditional probability of continuous independent random variables

I would like to know what's the exact way to obtain the conditional probabilities of node having multiple independent continuous random variables as its parent. Say something like a Noisy OR gate ...
2
votes
1answer
91 views

a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
0
votes
0answers
14 views

Trouble with Largrange Multipliers and expectation.

I am reading the following argument: Maximize $E[Log(X(T))]$ subject to $E[Z(T) X(T)]=x$, where $X(T),Z(T)$ are random variables, $x$ is a constant, and E is expected value (You can read this as ...
0
votes
0answers
24 views

Understanding certain parts of the proof of Helly's Selection Theorem

I have read through the following proof of Helly's Selection Theorem. There are just two parts, which I have highlighted, that are left for the reader to fill in, and I would like to know how to prove ...
1
vote
1answer
66 views

Is there closed form for $(1-p)(1-p^2)(1-p^3)…$ or its Taylor expansion?

I was considering the following problem: Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and ...
1
vote
1answer
65 views

Solutions to a stochastic birth-death-immigration process

A population is undergoing a birth-death-immigration process. That is, the population size can increase by virtue of birth and immigration, and can decrease by virtue of death. The birth rate is ...
3
votes
2answers
97 views

Problem with infinite product measures

Given some measurable space $\left(X,\mathcal{F}\right)$ and two probability measures $\mu$ and $\nu$ on this space one can define ...
0
votes
1answer
42 views

solve the functional equation

Let $\phi : R-> C $ (complex numbers) $\phi(0)=1$ $ \phi(-t) = \overline{\phi(t)} $ ( continuous and bounded) solve the functional equation: $Re \phi(t)= \phi(t) \overline{\phi(t)}$ This is all ...
0
votes
1answer
70 views

Show that if E is measurable set and f is continuous on E, then f(E) is measurable set

Please tell me how to prove or disprove it ! Show that if E is measurable set and f is continuous on E, then f(E) is measurable set
2
votes
1answer
56 views

Convergence in Probability of a Sequence of Exponential Random Variables

If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability. My original approach was the following: ...
0
votes
0answers
41 views

Convergence of eigenvectors/eigenvalues for infinite, non-negative, irreducible matrices with row sums less than $1$

I actually had two questions but decided to put them in separate posts for clarity. Suppose you have a matrix $M$ with infinitely many rows and columns, and a sequence of matrices $M_m$ with the ...
0
votes
1answer
40 views

Properties of logarithmic mean.

I have been studying the logarithmic mean for the last few days now. Could someone please help me with the following two questions? 1) We know that the log mean is in between the geometric mean and ...
1
vote
0answers
32 views

Linearization of exponential function with expectation

I ran into the following formula in my class: $$ E[\exp(X)]\approx \text{constant}+E[X]. $$ Since the linearization takes place at $X=0$, I don't understand why the constant in the formula is not ...
1
vote
1answer
27 views

Set Difference Probability [duplicate]

Here is the question: Prove that for every $\epsilon>0$ and every set $A\in\mathcal{B}(\mathbb{R}^{n})$ there is a compact set $K\subset A$ such that $P(A\setminus K)\leq\epsilon$. --I have ...
0
votes
1answer
80 views

The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
1
vote
1answer
46 views

$\mathbb L^1 +$ a.s. convergence of sequence $(X_n)$ does not imply $\sup(x_n)$ is integrable

I'm looking for a counterexample. The setting is this: Given an probability space $(\Omega,\mathbb{F},\mathbb{P}) $, I look for sequence of random variables $(X_n)_n$ and a random variable $X$, all in ...
2
votes
2answers
83 views

how to solve this integral in survival analysis

Let $T$ be a positive random variable, $S(t)=P(T\geq t)$. Prove that $$E[T]=\int^\infty_0 S(t)dt.$$ I have tried this unsuccessfully.
0
votes
0answers
60 views

A naive question about “random” probability distributions

So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable $X$ taking values in the interval $[-1,1]$. Then, say, ...
2
votes
1answer
44 views

Calculating the odds of winning a card game

So my friend made this game and I want to the odds of winning his game. So his game is basically I pay \$$1$ to draw $2$ cards from the deck and guess $2$ numbers and one suit. For each correct guess ...
0
votes
1answer
125 views

Infinitely Often vs Almost Always

Let $A_n$ be an event ($n \in \mathbb{N}$). Why is it that $x$ occuring for all but finitely many $n$ be a subset of $x$ occuring for infinitely many $n$? What if $x \in A_{2k}$, then $x$ will be ...
1
vote
0answers
61 views

Integral of the product of the Gamma function and the Confluent Hypergeometric function

This question was posted previously on stats.SE. The characteristic function of the Fisher$(1,\alpha)$ distribution is: $$C(t)=\frac{\Gamma \left(\frac{\alpha +1}{2}\right) ...
1
vote
2answers
121 views

Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
0
votes
1answer
72 views

Inequalities of the quantile function [closed]

I'm trying to rigorously prove the following inequalities involving the quantile function $Y(a) = \inf \{x \in\mathbb{R} : a \leq F(x)\}$ where $F$ is the distribution function: 1) $F(x) < a \iff ...
1
vote
0answers
68 views

Why does it exist?

I can show that $\lim_{a \rightarrow \infty} \int_{-\infty}^{a} (\cos(tu)-e^{-\frac{t^2}{2}})e^{-\frac{u^2}{2}}du$ converges to 0 but I am not sure why this implies the convergence of $$\lim_{b ...
1
vote
2answers
138 views

Probability that $\frac{n}{2}$ bins are empty [close]

A Bloom filter of length $n$ was built. I have only the first $\frac{n}{2}$ bits of this filter. How will the false positive probability change? For the whole Bloom filter, the false positive ...
4
votes
0answers
45 views

How much larger is the $\sigma$-algebra than the algebra in Caratheodory extension?

Given a 'measure' $\lambda$ on an algebra $\mathcal{A}$ of sets, Caratheodory gives a way to extend this $\lambda$ to a $\sigma$-algebra. The idea is we define an outer measure (on all subsets) ...
1
vote
1answer
49 views

IFF conditions for convergence in probability and almost surely

I am working on a bunch of problems in preparation for an exam in Probability Theory. I have come across two similar questions that I need some assistance with. Suppose we have a sequence of ...
1
vote
1answer
38 views

Definition of a random variable in the context of a hypergeometric distribution

We defined a random variable in a probability space $(\Omega, E, P)$ as a map $X: \Omega \rightarrow \mathbb{R}$. Unfortunately, I somehow have the impression that this term "random variable is used ...
0
votes
3answers
65 views

Fast way to do this well-known integral (gaussian-distribution)

I want to evaluate $$ \frac{1}{\sqrt{2 \pi } \sigma}\int_{-\infty}^{\infty} x^2e^{-\frac{(x-\mu)^2}{2\sigma ^2}}dx.$$ The problem is, I don't want to run into heavy calculations. Therefore, maybe ...