2
votes
0answers
40 views

weak convergence and composition

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued ...
1
vote
0answers
46 views

Bounds for sum of random variables

Let $A_1,...,A_M$ be random variables, not necessarily independent. For each one of them I know that $P( A_i \geq a )\leq B_i, \quad i=1,2,...,M$. How can I retrieve lower/upper bounds for ...
-1
votes
0answers
38 views

A basic question on characteristic function

Suppose I have two random variables $X$ and $Y$ for which characteristic functions are same. Let $F$ and $G$ be their distribution functions. I have to prove that $F$ and $G$ have the same set of ...
2
votes
1answer
39 views

Finding Random variables measurable

If [0,1] is our sample space and our sigma algebra is generated by all segments of the form [0,2^(-n)]. How can we describe the random variables measurable with respect to our sigma algebra? I'm ...
0
votes
0answers
19 views

Conditional Probability Question - on route availability

Hey Guys I am seemingly stumped with this question I have gotten involving conditional probability and routes Suppose route $A$ to $B$ is available 0.5 of the time An alternative route to B from A ...
1
vote
0answers
26 views

Stochastic domination by coupling

The following is a slightly streamlined version of Exercise 7.5 in Dubashi & Panconesi's "Concentration of Measure for the Analysis of Random Algorithms": Let $X$ and $Z$ be independent random ...
0
votes
1answer
26 views

Expected value vs values which happen with the biggest probability

If $X$ is a random variable from binomial distribution $Bin(n,p)$, then $$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$ where $p$ is the probability of one success. The expected value of random variable ...
1
vote
3answers
34 views

Independence of max and min of a set of random variables.

Suppose $X_1,\ldots,X_n$ are independent and identically distributed random variables with cdf $F_X(x)$. Define $U$ and $L$ as $U=\max\{ X_1, \ldots ,X_n\}$ and $L = \min\{X_1,\ldots,X_n\}$. Are $U$ ...
2
votes
2answers
47 views

Is this probability inequality always true?

For $n$ random variables $X_1$, $X_2$, $\dotsc$, and $X_n$. Is it always true that: $$\mathbb{P}\left[\sum_{k=1}^{n} X_k>a\right]\geq\mathbb{P}\left[\max\{X_1, X_2, \dotsc, X_n\}>a\right].$$ ...
0
votes
0answers
19 views

How to calculate this CDF?

Let suppose that we have three points in the euclidean plan $\mathbb{R}^2$ which are depicted inside a circle of radius $R$ as follow: $P_1=(D,0)$ (the center of the circle), $P_2=(0,0)$, and ...
1
vote
1answer
15 views

Number of storms in a rainy season

This is a follow-up to my previous question. Now instead of finding a probability I would like to now find the expectation too. I will restate the question and my solution below. I would appreciate if ...
0
votes
0answers
28 views

Order statistics of random variables

Let $\{I_1, I_2, \dotsc, I_N\}$ be $N$ i.i.d random variables. I know that the smallest orders statistics and the largest one are defined respectively as follow: $$I_{(1)}=\min(\{I_1, I_2, \dotsc, ...
1
vote
0answers
46 views

Almost sure convergence of a sum of independent exponential random variables?

I'm in difficult with this exercise... I hope someone can help me. Let $X_1,X_2,...$ be independent random variables, $X_n\sim \exp(\lambda_n)$, where $$0 < \lambda_n\rightarrow \lambda , \lambda ...
-2
votes
0answers
23 views

Help with random variable to found probabilty (PDF)

Stuck in this example to found (PDF) in many conditions
-2
votes
1answer
26 views

I stuck in binomial probability (PMF) with parameter n & p

I stuck in this example, but I have many trying
3
votes
1answer
43 views

Difference between density and distribution [in formal mathematical terms]

A similar question has been already asked but its not in mathematical framework and therefore seems to be different. According to definitions from the book that I am reading, a random variable and a ...
0
votes
1answer
30 views

Equivalence, identity of random variables

Suppose I have $X \sim \text{Uniform}(0,1)$ and $Y \sim \text{Uniform}(0,1)$ As we all know $X+Y$ is a triangular distribution. What of $X+X$? Surely this is uniformly distributed on the interval ...
3
votes
1answer
53 views

Function of a uniformly distributed continuous random variable

Basically, I'd like to add $n$ random vectors in a 2 dimensional space of unit length and of angle $\theta$ relative to a global axis. The probability density function of the angle $\theta$ is a ...
0
votes
1answer
24 views

Convergence in probability of sample variance

$X_n$ s are a sequence off iid random variables with E($X_n$) = $\mu$, V($X_n$)= $\sigma$$^2$ and $\bar X = \sum$ $\frac{X_i}{n}$. Then show that $\frac1n$ $\sum (X_i - \bar X )^2\to\sigma^2$ in ...
2
votes
1answer
39 views

Product of $n$ i.i.d. random variables

Let the variable $Z$ equal $Z = XY$ where $X$ and $X$ are two i.i.d. continuous random variables which distributions are given by $f_X()$ and $f_Y$. The distribution of $Z$ is given by: $$f_Z(z) = ...
0
votes
1answer
29 views

What is the distribution of the dot product of a Dirichlet vector with a fixed vector?

I am trying to get the distribution of a weighted sum when the weights are uncertain: $S = \sum\limits_{i=1}^N w_iC_i = \mathbf{w}\cdot \mathbf{C}$ where vector $\mathbf{w}$ is random with components ...
0
votes
1answer
21 views

Condition for independence of two scalar real valued random variables

I'm trying to show that given two real-valued scalar random variables $X,Y$ if $$\mathbb{P}\left(X\leq x,\, Y\leq y\right)\cdot\mathbb{P}\left(X\geq x,Y\geq y\right)=\mathbb{P}\left(X\geq x,Y\leq ...
0
votes
1answer
33 views

Given the distribution of a random variable $R$, who do you get a uniform random variable $U$?

Let us say you have a random variable $R$. How would one generate a uniform random variable $U$, with the maximum possible entropy (or infinite entropy, if $R$ has such)? (For simplicity, you may ...
5
votes
0answers
56 views

Covergenge of the sum of reciprocal random variable.

If $(X_n)_{n\in\mathbb{N}}$ are independent identically distributed random variables with density $f$ even, continuous in $0$ and such that $f(0)>0$, then $$\frac{1}{n}\left(\frac{1}{X_1}+\dots + ...
1
vote
1answer
42 views

When does distribution convergence imply expectation convergence?

If $X_n \xrightarrow{d} X$ what are the minimal hypothesis to have $E[X_n]\rightarrow E[X]$ ? For example I think that if all the second moments are bounded it's true, but I'm not sure if is true if ...
0
votes
0answers
23 views

Looking for a distribution where the scale amd sum lead to a closed form distribution

As the title says, say I have a finite number of i.i.d. random variables with positive support $X_1,X_2\dots,X_n$. I'm interested in finding a closed form expression for $S$, where: $S=\sum a_i X_i$ ...
3
votes
0answers
84 views

Skorohod representation theorem

Assume $X_n$ are random variables such that $\mathbb{P}(X_n \leq B)=1$ for some random variable $B$. Assume also $X_n \Rightarrow X$ ($X_n$ converges weakly to $X$). By the skorohod representation ...
1
vote
1answer
32 views

A basic problem on random series/ law of large numbers

Consider the following two statements : i) Suppose that $X_1, X_2, \dots$ are independent and identically distributed and $E[X_1^-] < \infty, E[X_1^+] = \infty$. Then $n^{-1} \sum_{k=1}^{n}X_k ...
0
votes
0answers
14 views

Calculating the joint distribution of an affine stochastic process

I have a recursively defined system given by $$X_i = X_{i-1}H_i+N,$$ where $H_i$s are i.i.d. exponential random variables and N is a constant. At the $n$th iteration I have $$X_n = ...
1
vote
2answers
39 views

Product & Ratio's of 2 Random Variables

I'm interested to know whether it's the case that for random variables $X$ and $Y$ whether or not the ratio of $X$ and $Y$ can be computed as the product of $X$ and $1/Y$. That is, Is $\frac{X}{Y} ...
1
vote
0answers
21 views

Prove that the empirical measure is a measurable fucntion

This problem came from Schervish, Theory of Statistics, Sec. 1.4 Prob. 24. Suppose that $X_1, \ldots, X_n$ are exchangeable and take values in the Borel space $(\mathcal{X}, \mathcal{B})$. Prove ...
2
votes
0answers
68 views

A nice sequence of random variables

Let $f:U\mapsto \mathbb{R}^k$ with $U\subset \mathbb{R}$ be a smooth injective function. Suppose that $\sqrt n(Y_n- Y)\to N(0,\Omega)$ in distribution with $Y=f(X)$. Define $X_n$ by ...
3
votes
3answers
81 views

Two random variables from the same probability density function: how can they be different?

The definition of $X$ as a random variable according to Wiki is as follows: $Let (\Omega, \mathcal{F}, P)$ be a probability space and $(E, > \mathcal{E})$ a measurable space. Then an $(E, ...
0
votes
2answers
61 views

Is a non-negative random variable with zero mean almost surely zero?

We have proven the following in class: If $X$ is a finite random variable with $X\geq 0$ then $$E(X)=0 \iff P(X=0)=1$$ (By finite I meant that the range has finitely many elements). Does it ...
0
votes
1answer
31 views

Bounded function of geometric random variable

if X~ Geometric(p), with q=1-p, then show that for any bounded function f with f(0)=0, we have E(f(x)-qf(x)+1)]=0. Our professor asked us to try solving this problem as a good practice but I have no ...
1
vote
1answer
23 views

A mapping that does not preserve convergence in distribution

I'm trying to come up with a map $H: \mathbb{R}^k\to \mathbb{R}^k$ and a sequence of random vectors $X_n\Rightarrow X$ in $\mathbb{R}^k$ for which $H(X_n)$ does not converge in distribution to $H(X)$. ...
0
votes
2answers
75 views

Conditional expectation of number of dice rolls

I've been given the following problem and I'd like to get a better understanding of how to solve it. A fair die is rolled successively. Let $X$ be the number of rolls needed to get a 6 Let $Y$ be the ...
1
vote
1answer
34 views

Show: $X_n\xrightarrow{\mathcal{d}} X$, then $\mathbb{E}\lvert X\rvert\leqslant\liminf_{n\to\infty}\mathbb{E}\lvert X_n\rvert$

Let $X_n, X$ be random variables with $X_n\xrightarrow{d} X$. Show that then $$ \mathbb{E}\lvert X\rvert\leqslant\liminf_n \mathbb{E}\lvert X_n\rvert. $$ So let $X_n\xrightarrow{d} ...
0
votes
2answers
65 views

Formal definition of a random variable

I'm not new to the concept of random variable and I know the measure theory. Anyway, I started reading the book "Stochastic Differential Equation" by B. Oksendal, and I'm having some problem in ...
0
votes
1answer
33 views

$(\dots, X_{t-1}, X_t)$ and $(X_{t+1}, X_{t+2}, \dots) $ are independent $\Rightarrow (X_t)$ are independent

In some lecture notes in Time Series Analysis it was written: If the random vectors $(\dots, X_{t-1}, X_t)$ and $(X_{t+1}, X_{t+2}, \dots)$ are independent for all $t \in \mathbb{Z}$, then it is ...
0
votes
0answers
22 views

Density of Gaussian Unitary Ensemble

I'm trying to learn a bit about Gaussian matrix ensembles, and am having some trouble making the following connection. Sorry if I'm being a bit obtuse. Take the Gaussian unitary ensemble (GUE) of $n ...
0
votes
2answers
43 views

Can any sequence of RVs converge almost surely to a non-constant RV?

We know for almost sure convergence, $X_n \rightarrow X \text{ a.s. as } n\rightarrow \infty$ if $$ P\{\limsup_{n\rightarrow\infty}X_n = \liminf_{n\rightarrow\infty}X_n\} = 1. $$ Then does this mean ...
2
votes
0answers
81 views

Coupling Pairs of Random Variable.

Let $\{X_i\}_{i=1}^{n}$ and $\{Z_i\}_{i=1}^{n}$ be sets of independent random variables with coupling $\{X^{\hat{}}_i\}_{i=1}^{n}$, $\{Z^{\hat{}}_i\}_{i=1}^{n}$ respectively. It then states ...
0
votes
1answer
34 views

Prove that ${(1-p+pe^t)}^n = E(e^{t\sum Z_i}))$

Let $Z_1,...,Z_n$ be independent random variables such that $ Z_i \in \vert0,1\vert$. Prove that ${(1-p+pe^t)}^n = E(e^{t\sum Z_i}))$ where $p=\sum{\frac{E(Z_i)}{n}}$ Not quite sure how to do this. ...
1
vote
0answers
46 views

If $X$ is a random variable, under which conditions is $g(X)$ also a r.v.?

In many instances, functions of random variables appear, and we usually treat them as random variables also. In the 3d edition, pp. 85-86, of this well-known book (now in its 4th edition), we find the ...
2
votes
1answer
31 views

A basic doubt on sigma algebra generated by a random variable

Why do we need the concept of sigma algebra generated by a random variable ?
4
votes
2answers
189 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
0
votes
0answers
13 views

Robbins-Siegmund like theorem for a nonlinear system

Recently I came to know about this theorem due to Robbins and Siegmund which states the following: Let us have on a probability space $(\Omega, \mathcal{F},P)$, a filtration $\{\mathcal{F}_n\}$ and ...
0
votes
1answer
27 views

Use of convolutions to compute the distribution of the sample mean?

Let's consider N i.i.d continuous random variables from some arbitrary distribution. Why do we have to approximate the distribution of the sample mean using the CLT? Why can't we explicitly compute ...
3
votes
1answer
83 views

Proving increasing function defined as bivariate normal

Suppose $c>0,\sigma>0$ and $\tau>0$ are fixed real constants. Then I'd like to prove that the function $g_c:(-1,1)\mapsto\mathbb{R}$ defined by \begin{equation} ...