3
votes
2answers
68 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
-2
votes
1answer
33 views

Mean of max vs max of mean

If I have say an $n$ collection of 10 random variables $X_1, \ldots, X_{10}$ (so an $n \times 10$ matrix of values) from some underlying distribution whether Gaussian or uniform, and I calculate ...
1
vote
1answer
19 views

Does negative part of a standardized random variable converge to negative part of a $\mathcal{N}(0,1)$?

I know how to prove that any standardized random variable converge in distribution to a $\mathcal{N}(0,1)$, I was wondering if even $f((S_n-n)/ \sqrt{n}))$ converge to $f(\mathcal{N}(0,1))$, in ...
0
votes
0answers
23 views

Testing for the independence of random variables

In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$ If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I ...
1
vote
1answer
26 views

Why is $\limsup\limits_{n\to\infty}X_n$, $C_{\infty}$-measurable?

Why is $\limsup\limits_{n\to\infty}X_n$, $C_{\infty}$-measurable ? If $\mathcal B_n=\sigma(X_n)$,$\quad$$\mathcal C_n=\sigma\left(\bigcup_{m\ge n}\mathcal B_n\right)$,$\quad$$\mathcal ...
0
votes
2answers
43 views

Existence of density function for a sum of 2 Random Variables

Let's suppose that $Y$ is the normal distribution and that $X$ is another random variable whose density function may or may not exist. Does it follow that $Y+X$ has a density function? I am reading ...
2
votes
2answers
72 views

$X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P) $ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
1
vote
1answer
32 views

Measurability of a Borel function

I need some help on the following proof. The claim is: Suppose $f:\mathbb{R}^k \to \mathbb{R}$ and $f \in B(\mathbb{R}^k)/B(\mathbb{R})$. i.e. Borel measurable. Let $X_1$,...,$X_k$ be random ...
1
vote
0answers
48 views

Problem regarding Conditional probability

Let $\mathbf{X}$ be an $n-$ dimensional random variable. This variable can be written as $\mathbf{X} = \left[\mathbf{X}_1^T\hspace{5pt}\mathbf{X}_2^T\right]^T$. where, $\mathbf{X}_1$ is $m-$ ...
1
vote
0answers
51 views

Azuma's inequality: basic question [closed]

In the statement of Azuma's inequality, it is assumed that random variables $X_n$ are martingales (and therefore $X_n - X_{n-1}$ are martingale differences). But what is the essential step in the ...
1
vote
0answers
53 views

Prove that a.s.$\lim\limits_{t\to\infty}\frac{N_t}{t}=\frac{1}{\mu}$

Consider a diligent janitor who replaces a light bulb the instant it burns out. Suppose that the first bulb is put in at time zero and let $X_i$ be the lifetime of the i-th bulb. Suppose ...
2
votes
1answer
47 views

random variables not independent but $\mathrm{E}[X|Y]=\mathrm{E}[X]$

I have to find two r.v. X,Y defined in a probability space ($\Omega, \mathcal{F}, \mathrm{P}$), which are not independent but for which $\mathrm{E}[X|Y]=\mathrm{E}[X]$ nonetheless, with ...
1
vote
1answer
32 views

Random variable bounded by another random variable

How to find $\Pr(z<X<Y)$ if $X$ and $Y$ are independent exponential r.v.'s with parameters $\lambda$ and $\mu$ So $x$ is bounded by $z$ and $y$, and y must be go from $z$, (and not from ...
3
votes
1answer
47 views

Proving independence of random variables

If $X$ and $Y$ are independent exponential random variables with parameter $\lambda$ and $\mu$. Let $Z=\min(X,Y)$, prove that $Z$ and $\mathbf 1_{\{X<Y\}}$ are independent. I don't know, how ...
0
votes
1answer
26 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
1
vote
3answers
33 views

Is $E[Bin(X,p)]=E[X]p$?

We have given some random variable $X$ with mean $E[X]=:\mu$. Now we are interested in a random variable $Y \sim Bin(X,p)$. Is it true that $$E[Y]=\mu p?$$ What confuses me is that normally the ...
6
votes
0answers
70 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...
2
votes
1answer
64 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
0
votes
0answers
23 views

For a sequence of random variables with bounded probability density function, can their joint pdf be unbounded?

For $d\geq2$, let $X_{i}=\left\{Y_{i-1},Y_{i-2},...,Y_{i-d} \right\}$, and assume the sequence $\left\{X_i \right\}$ is strictly stationary. Let $f_{j}(x_{0},x_j)$ denote the joint density of ...
3
votes
1answer
62 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
3
votes
1answer
71 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
0
votes
0answers
48 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
1
vote
0answers
15 views

Are functions of independent random variables related to each other by a constant independent

I have $6$ random variables $a,b,c,d,f,g$, each having independent Gaussian distribution. Now I define following three random variables \begin{equation} X = ab - cd\\ Y = cf - ag\\ Z = gd - bf ...
1
vote
0answers
30 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
1
vote
1answer
26 views

Product of 2 random variables:domain of integration

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$) I found this formula : $ f_Z(z) = ...
1
vote
2answers
43 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
3
votes
1answer
90 views

If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}] $?

The question itself is in the title. It is immediate by the strong law of large numbers that if $X_{i}$ had a finite first moment then we would have a.e convergence (and thus in probability and in ...
6
votes
1answer
54 views

Finding tight upper/lower bounds for $\mathbb{E}[\frac{1}{1+X^{2}}]$ where $X$ is a RV with $\mathbb{E}[X]=0$ and $\mbox{Var}(X)=\nu<\infty $

The question is pretty much in the title. My first thought was using Jensen's inquality to get some sort of lower bound. Since $\frac{1}{1+x^{2}}$ is convex on ...
7
votes
0answers
62 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
0
votes
1answer
31 views

Show that $\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$

Let $A=[-\frac1u,\frac1u]$, Show that $$\displaystyle\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$$ where $\Phi_X(u)$ is the characteristic function of the r.v. $X$ Hint: ...
0
votes
1answer
40 views

Show that $\Pr(S_N\in A\mid N=n)=\Pr(S_n\in A)$

Let $X_1,.\ldots,X_n$ be i.i.d. random variables and $N$ be a positive integer-valued random variable, which is independent from the sequence. If $S_n=\displaystyle\sum\limits_{i=1}^{n} X_i$, then ...
0
votes
1answer
18 views

How to find the LimInf of a sequence of RV's

I have an indipendent sequence of Rv's $Y_n$ with law that eventually is: $\mathbb{P}(Y_n<a) = \begin{cases} 0 & \mbox{if}\ a \leq 0 \\ 1 - \frac{1}{n^{2\gamma}a^2}& \mbox{if}\ a ...
2
votes
0answers
20 views

Density of the $k^{th}$ smallest of $X_1,X_2,…,X_n$

Show that if $(X_1,X_2,...,X_n)$ are i.i.d. with common density $f$ and distribution function $F$, then $X_{(k)}$ has density $$f_{(k)}=k\binom{n}{k}f(y)(1-F(y))^{n-k}F(y)^{k-1}$$ where ...
0
votes
1answer
26 views

Evaluating an expectation of the supremum of collection of random variables

I know that $\mathbb{E}(sup_n|X_n|)=\infty$ if $X_n=\frac{2^n}n\cdot\mathbf 1_{(1/2^{n+1},1/2^n)}$. However I am not sure how this can be evaluated explicitly. The probability space is $Ω=[0,1]$ ...
1
vote
1answer
48 views

First hitting time expectation and Markov property

Let $H_A$ be the first hitting time, such that $H_A\geqslant1$, so we have $X_0=i\notin A$. All texts I looked at, state without any further justification that $$ \mathbb E(H_A\mid X_1=j, ...
3
votes
1answer
35 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
1
vote
2answers
96 views

Tossing a coin with at least $k$ consecutive heads

Toss a coin with $\Pr(\text{Heads})=p$ repeatedly. Let $A_k$ be the event that $k$ or more consecutive heads occurs amongst the tosses numbered $2^k,2^k+1,...,2^{k+1}-1$. Show that, $\Pr(A_k\ ...
0
votes
1answer
73 views

Is this true: probability independent from i?

We have a set of i.i.d. random variable $X_i$ with some discrete distribution. Further we have a random variable Y, Independent from $X_i$ with a Binomial Distribution Bin(n,p). Now we are ...
1
vote
1answer
33 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle ...
2
votes
1answer
21 views

Modes of Convergence of a particular random variable

Let $X_n \sim U([-1/n,1/n])$ be uniform random variables on $[-1/n,1/n]$ for $n \in \mathbb{N}$. Do the $X_n$ converge, and if yes in what sense? I think it converges pointwise as for any $x \in ...
0
votes
0answers
37 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
2
votes
1answer
44 views

Measures in conditional expectation.

I always make confusion when a measure has to be changed in some other measure. This time I'm stuck on a change of measure in the definition of conditional expectation of a random variable. If $Z$ is ...
0
votes
1answer
33 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
1
vote
0answers
30 views

If $\{X_n\}$ is a Cauchy seq of r.vs a.s., then why does it converge to some r.v?

I was suddenly suspicious of what title says. My logics are follows. $\{X_n\}$ is a Cauchy seq of r.vs $P$-a.s. $\Leftrightarrow$ $P(\{X_n\}$is a Cauchy $)=1$ $\Leftrightarrow$ $\exists ...
2
votes
1answer
68 views

Proving it's a martingale and more conditions.

Let $(X_{n})_{n>0}$ be a sequence of random variables in $[0, 1]$ and assuming that ($X_{0}=a) \epsilon [0, 1]$ then: $Pr\left(X_{n+1}=\frac{X_{n}}{2}|\mathcal{F}_{n}\right)=1-X_{n}$ and ...
0
votes
0answers
39 views

How to show $ P\big(\big|\frac{X}{n}-p\big|>a\big)\le\frac{\sqrt{p(1-p)}}{a^2n}min\big\{\sqrt{p(1-p)},a\sqrt{n}\big\}$

Let $X$ be binomial, $B(p,n)$ with $p>0$ fixed, and $a>0$. Show that, $\displaystyle ...
1
vote
3answers
70 views

Expected value of the function of a random variable

I am studying Probability and Monte Carlo methods, and it feels that the more I study the less I truly understand the theory. I guess I just confuse myself now. So the expected value of a random ...
1
vote
1answer
29 views

Showing independence of rectangular events…

Suppose I have a sequence of independent random variables $\{X_n, n \in \mathbb N\}$. How do I show formally that $P((X_1,...,X_n)\in A, (X_{n+1},...)\in B) = P((X_1,...,X_n)\in A)P((X_{n+1},...)\in ...
2
votes
1answer
32 views

Convergence almost surely and B-C lemmas

Showing the expectation is straightforward. I am not sure how to use the Borel-Cantelli lemmas to show the almost surely part.
1
vote
0answers
26 views

Random sampling and i.i.d.

Can you help me to clarify the following concepts by stating whether what I have written below is right or wrong? -random sampling: units are drawn from the population with a known probability of ...