0
votes
0answers
27 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
18 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
16 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
33 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
23 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
23 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
50 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
34 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
75 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
49 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
41 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
94 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
47 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
115 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
51 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
67 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
133 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
33 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
42 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
36 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
57 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 ...
0
votes
0answers
38 views

PDFs with a specific Coefficient of Variation ($\sigma/\mu$)

So, I am trying to find the general expression for the probability density functions that have a specific Coefficient of Variation. The Coefficient of Variation is the ratio between $\sigma$ (the ...
0
votes
0answers
72 views

Probablity - A question about Poisson distribution and Stirlings Formula leading up to the central limit theorem

The goal of this problem is to prove that $P(S_n = k)\sqrt{2\pi n} \rightarrow e^{-\frac{x^2}{2}}$ by using Stirling's formula. Here is what is given: 1) $S_n = \sum_1^n X_i$, where $\{X_i\}$ are ...
0
votes
1answer
57 views

log partition function of exponential family

In an exponential family $$p_{\theta}(x)=\exp \left(h(x)+\sum\limits_{i=1}^s \theta_iT_i(x) - \phi(\theta) \right) $$ is the log partition function $$ \phi(\theta)=\log \int \exp ...
0
votes
0answers
36 views

Chernoff bound in binomial case

Let $S_n$ the binomial distribution with parametres $n,p$. I have to prove that $$P(S_n\geq n(p+\varepsilon))\leq e^{-2n\varepsilon^2}$$ for every $\varepsilon\geq 0$. I have to use Stirling's ...
0
votes
1answer
32 views

Raffle Odds and Payout

There are four raffles. One raffle ticket costs $\$500$. Raffle one sells $500$ tickets with winning ticket a payout of $\$80,000$. Second raffle one raffle ticket costs $\$150$ only sells $1,500$ ...
0
votes
1answer
31 views

Why would $f_n(x) = (\lfloor 2^nf(x)\rfloor/2^n)\wedge n$ converge to $f(x)$?

Why would $$f_n(x)=\frac{\lfloor 2^nf(x)\rfloor}{2^n}\land n$$ converge to $f(x)$? I saw this step in the proof of change of variable formula in Rick Durrett's Probability Theory and Examples.
1
vote
1answer
57 views

Many points on hyperplane with probability zero

Let $m$ be a finite measure on $X \subseteq \mathbb{R}^n$, so that $m(\mathbb{R}^n) < \infty$. Define the hyperplanes on $\mathbb{R}^n$, parametrized by $A \in \mathbb{R}^{n \times n}$ and $b \in ...
3
votes
1answer
124 views

definition of “weak convergence in $L^1$”

I have encountered two definitions of weak convergence in $L^1$: 1) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n\mathrm{1}_A)\rightarrow \mathrm{E}(X\mathrm{1}_A)$ for every measurable set ...
5
votes
1answer
227 views

Jensen's Inequality (with probability one)

In the following theorem, I have a problem about the second part. That is showing if $f$ is strictly convex then $X=EX$ with probability $1$. While I can see this must be true, I don't know how to ...
0
votes
2answers
104 views

Is there a random variable where E[X] exists but the expectation of the negative and positive part do not?

Is there a random variable where $E[X]$ exists but the expectation $E[X^{+}]$ and $E[X^{-}]$ do not? If not, how can I show this? I know that it is usually written that $$ X = X^{+} - X^{-}$$ and ...
2
votes
1answer
53 views

Is this inequality true? (Inequality involving probability distribution and products)

Suppose $f(z)$ is a discrete probability distribution with space $S$. Suppose $g(z),h(z)>0$ for all $z \in S$. Is it true that $$\prod_{z \in S}{g(z)^{f(z)}}+\prod_{z \in S}{h(z)^{f(z)}} \leq ...
1
vote
1answer
76 views

Ratio of convex functions with dominating derivatives is convex?

Let $f,g:\mathbb [0,\infty)\rightarrow (0,\infty)$ satisfy $f^{(n)}(x)\geq g^{(n)}(x)>0$ for all $n=0,1,2,\ldots$ and $x\in [0,\infty)$. In particular, $f\geq g> 0$ are increasing and convex ...
2
votes
2answers
176 views

Almost sure convergence of a sum of random variables

Suppose $(X_i)_{i=1}^{\infty}$ is an i.i.d. sequence of rv's, where $X_i$ can take countably many values $\{x_1,x_2,\dots\}$ with probabilities $\{p_1,p_2\dots\}$, respectively. Let $p_{n,k}:= ...
1
vote
1answer
98 views

minimization of L^2 norm of the second derivative of a probability density

I have a question: Let $\rho$ denote a probability density function defined on $[0,1]$. It is twice-differentiable and has a continuous second derivative. Denote by $M$ the set of all such functions ...
7
votes
1answer
142 views

Prove the density of this SDE is not smooth in a parameter

Consider the following, 1-dimensional, equation $$X_t^x = x + \int_0^t \mathbb{E} |X_s^x| \, ds + B_t , $$ where $B$ is a Brownian motion. This a McKean-Vlasov equation, sometimes called a nonlinear ...
4
votes
1answer
96 views

How can I approximate $\sum\limits_{k=4}^{\infty}\Pr(X=k)[{\Pr(X\le k)}^6 - {\Pr(X\le k-4)}^6]$ for $\lambda \to +\infty$?

$X$ is a Poisson random variable and the probability mass function is given by: $$\Pr(X = k) = e^{-\lambda}\frac{{\lambda}^k}{k!}$$ I’ve got a probability function $f(\lambda)$ $$f(\lambda) = ...
3
votes
2answers
52 views

Probability Calculation using combinations

In a population of $250$ items, $20$ are defective. Suppose $4$ items are sampled at random, without replacement. a. What is the probability that the sample will consist of $4$ defective items? ...
1
vote
1answer
65 views

Estimate the scale of the power series with Poisson pdf-like terms

Sorry to bother you, but I guess that this question is not appropriate for MO, so I repost it here hoping that someone could give me a clue. I would like to have an estimate for the series $$P(t) = ...
0
votes
0answers
36 views

Comparison of Probabilities of Getting a Formula in Different Notational Schemes

I played the first two WFF 'N Proof games of the WFF 'N Proof kit tonight with a friend, and on my way home I started thinking. Suppose we have the set of variables, constants, and truth-functions ...
0
votes
1answer
127 views

Examples of convergence of random variables

First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution: $X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
1
vote
1answer
159 views

Find $c$ such that $\text{P}(\limsup\limits_{n\to \infty} X_n/\sqrt{\log n}=c)= 1$

Need to find $c$ such that $\text{P}(\limsup\limits_{n\to \infty} X_n/\sqrt{\log n}=c)= 1$, where $X_n$ are a sequence of independent random variables such that $X_k\sim\mathcal{N}(0,1) $ I need to ...
0
votes
0answers
32 views

Gap distribution independence proof

I have a question bout the proof of the independence of gap RVs. Given the independent exponentially distributed random variables $\xi_1$, $\xi_2$ ~ $\text{Exp}(\lambda)$, and a corresponding order ...
4
votes
1answer
216 views

Proof of Khintchine's inequality

I'm trying to understand the proof of Khintchine's inequality in these lecture notes: http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf On page 27, second display-style equation after (51), the ...
1
vote
2answers
187 views

Derive the probability density function of the distance

I was and still wondering how to derive the probability density function of $d^2$, where $d$ is the distance between two points that are inside a circle of radius $R$. The two points are uniformly ...
0
votes
1answer
332 views

Uniform sum distribution

I was wondering how to derive the probability density function for the sum of $n$ independent iid distributed random variables on the interval $[0,1]$. A formula for that is given on ...
2
votes
1answer
101 views

Equivalence of measures and $L^1$ functions

Suppose we have two probability measures $\mu$ and $\delta$ on $(X, \mathcal{B})$ such that $ \delta <<\mu << \delta $. How can I prove that $f \in L^1(X,\mathcal{B}, \mu)$ iff $f \in ...
2
votes
1answer
251 views

Proving that a function is negligible

In mathematics, a negligible function is a function $\mu(x):\mathbb{N}{\rightarrow}\mathbb{R}$ such that for every positive integer $c$ there exists an integer $N_c$ such that for all $x > N_c$, ...
1
vote
1answer
333 views

Difficult and unusual probability problem, how to solve?

Let $n_i$ be the $i$'th randomly chosen element from $\mathbb N$ with replacements. All elements have probability greater than zero of being chosen. After a number of trials $k$, the probability ...
0
votes
1answer
329 views

Expected hitting time of one of two barriers

In the webpage "hitting time of one of two barriers", the probability that a non symmetric random walk hits one of two barriers is computed. The walker starts from $x=0$ and the barriers are located ...