Questions about maps from a probability space to a measure space which are measurable.

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0answers
22 views

Convergence of discrete random variables, show $\displaystyle\frac{S_n}{\sqrt{n}}\to0$ a.s.

Let $X_n$ be a sequence of independent discrete real random variables, with discrete density $$p_{X_n}(x):=\Pr(X_n=x)= \cases{ 1-\frac1{n^2} & \text{if } x= 0\cr \frac1{2n^2} & \text{if ...
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1answer
20 views

Expected length of a random vector

I meet a basic definition about the expected length of a random vector when reading a paper: What is "expected length" How to roughly derive both equations (yellow part) (Is that Gamma ...
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1answer
22 views

Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...
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1answer
48 views

What is the mean and variance of $Y$, where $Y$ is sum of iid's

Here's my work for part a. I could use clarification on part b and d. Is part d the same as part a ($E[A_n] = E[Y]$) ? a) $$E[Y_n] = E[\frac{X_n}{2^n}]$$ ($X$'s are iid so...) $$= \frac{E[X]}{2^n} ...
2
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1answer
16 views

Why $E[X|\mathcal{G}]=X$ if $X$ is $\mathcal{G}$-measurable?

If $X$ is a $\mathcal{G}$-measurable random variable, why $E[X|\mathcal{G}] = X$? I know the intuition (basicly we're conditioning on the same informations on which $X$ is defined, $\sigma(X)$, we ...
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2answers
43 views

For random variables, show that $\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$

Why is the following true ? $$\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$$ where, $X_n's$ are random variables. If we consider only finitely many $X_n$, say ...
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2answers
21 views

Compensation Question

I want to create a compensation system which takes into account two variables. Lets say I have $1M to distribute among ten employees who produce widgets. I want to compensate each employee by two ...
0
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1answer
38 views

Convergence sequence of random variables

I have this problem about a sequence of normals. $(X_n)_{n\geq 0}$ is defined as $$X_{n+1}=aX_n+U_{n+1}$$ $X_0=0$, where $(U_n)_{n\geq1}$ is a sequence of i.i.d random variable normally distributed ...
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1answer
55 views
+50

Mean of Piecewise function resting on IID random variables

Suppose IID random variables $X_t \sim X$ with support on $[0,1]$ and continuous CDF $F(\cdot)$. I wish to compute the expected value (mean) of the a piecewise function with form $$ \Phi (x,\mu) = ...
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3answers
42 views

Transformation(?) of Random Variables

There are two independent Gaussian R.Vs: $U:N(-1,1)$ and $V:N(1,1)$ How do I go about finding the PDF of the following transformations? X = U+V T = (U+2V, U-2V) W = U (with 50% chance), V (with ...
2
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1answer
13 views

Distribution of a function of a normally distributed variable

Let's say you have a random variable $X$, which is normally distributed according to $X \sim \mathcal{N}(1,2)$. With $1$ being the mean and $2$ being the variance. Now let's say that there is another ...
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0answers
20 views

Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
3
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2answers
80 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} ...
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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 ...
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1answer
34 views

Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely

I'm trying to solve the following Problem: Let $(X_n)_{n\ge 1}$ be a sequence of real valued random variables defined on some probability space $(\Omega, \mathcal{A},P)$. Assume that there ...
1
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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 ...
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0answers
24 views

About independent random variables

Suppose that $\{X_n\}_{n\in\mathbb N}$ are identical distribued and independent random variables with values in $\mathbb Z$. I don't understand why ...
5
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0answers
50 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
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0answers
24 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 ...
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2answers
29 views

Identifying the distribution which represents a negative binomial distribution as a compound poisson distribution

Suppose that the random variable $X$, which has a negative binomial distribution with probability $p$ and parameter $r$, can be represented as the summation of $N$ iid random variables $Y_1, Y_2, ...
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0answers
29 views

Probability distribution of k consecutive successes with n maximum trials

Let $X$ be a random variable that represents the number of trials of a given experiment. The outcome of a single trial is a Bernoulli random variable, with probability of success $p$, and trials are ...
0
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1answer
42 views

Expected value and variance of random process

Let $U,V$ be random variables with distributions $\mathcal{U}(-1,1)$ ,$\mathcal{E}(2)$ (uniform and exponential). If $U$ and $V$ are independent what is the variance and expectation of the random ...
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3answers
53 views

Are the absolute values of random variables iid if the random variables are iid?

If $X$ and $Y$ are independent and identically distributed (iid) random variables, does it imply that $|X|$ and $|Y|$ are iid? How would you go about proving this?
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1answer
15 views

What is the variance of multiple indicator random variables?!

Consider the following independent random variables $(V_1,V_2,V_3,\ldots,V_n)$ and a random variable $X$ as a function of these other random variables defined as follow on a set $A=(-\infty,x]$: $$ \ ...
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2answers
32 views

Marginal distributions of a random vector

I have the random vector $(X,Y)$ with density function $8x^{2}y$ for $0 < x < 1$, $0 < y < \sqrt{x}$ I am trying to find the marginal distributions of $X$ and $Y$. For $X$ this seems to be ...
0
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0answers
38 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
1
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1answer
32 views

Bolzano–Weierstrass theorem for random variables?

I am wondering if there is something similar to the Bolzano–Weierstrass theorem for random sequences. Namely, let $\{x_n\}$ be a bounded random sequence. Is it true that, under some reasonable ...
0
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1answer
18 views

Show $P(X=n)=\left(\frac{1}{2}\right)^{n+1}$ for Poisson variable with exponentially distributed $\lambda$

I'm supposed to do the following, any help/pointer is appreciated: Suppose $X$ is Poisson distributed with mean $\lambda$. Suppose $\lambda$ is exponentially distributed with mean $1$. Show that ...
0
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1answer
39 views

How to increase winning chance in lottery [closed]

Let us imagine such kind of lottery game :lottery machine is running and randomly is selecting $7$ number from $1$ to $36$(including).out of this $7$ numbers,$6$ are basic or in other word ,jackpot ...
1
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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 ...
1
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1answer
33 views

Distribution of number of Poisson arrivals in interval

$X_1$ and $X_2$ are both Poisson processes. $N$ is the number of arrivals of $X_1$ in between two subsequent arrivals of $X_2$. Derive the probability density $f_N(n)$ of $N$. I wanted to start from ...
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1answer
32 views

Covariance of a function of random variables

I want to find the covariance $K_X(t,t')$ of the following signal $X(t)$: $X(t)=\sum\limits_{n=-\infty}^{+\infty} A_np(t-nT)$ where $ p(t) = \begin{cases} \ 1 & \text{if } 0<t\leq T/2 ...
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2answers
44 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 ...
1
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2answers
36 views

Probability distribution and density of Y=g(X)

$Y=g(X)$ as shown below. Find $f_Y(y)$ and $F_Y(y)$ in function of $f_X(x)$. I began with writing $Y=g(X)$ as the following piecewise function: $ Y = \begin{cases} \ -b & \text{if } x ...
2
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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 ...
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0answers
45 views

An expectation inequality [on hold]

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ...
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2answers
40 views

Question about computing expected value of the limit of a geometric mean of random variables

If I have the random variables $ X_{i} $ for $ i=1 \ldots N$ with the random variables being randomly selected integers from $1$ to $9$, how would I calculate the expected value of $$\lim_{N \to ...
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0answers
19 views

Probability Density Function for Discrete and Continuous Random Variables.

From the lecture on Khan's Academy. While discussing the Discrete Random Variable, consider the following RV P[X=0] = 1/4; P[X=1] = 1/2; P[X=2] = 1/4; And after ...
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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 ...
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0answers
23 views

Conditional return time of simple random walk

Consider a simple random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0 = 0$. The probability to jump to the right neighbour is $p \geq \frac{1}{2}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, ...
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0answers
25 views

probability that a variable, as a function of choice variables, is among the top k out of n when ordered

Suppose $(h_1,h_2,...,h_n)'$ is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a choice variable which can vary across $i$ and $X_i$ is a random shock with Pareto Type I distribution. ...
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1answer
35 views

Odds to guess a 32 byte value [closed]

I have 1,000,000 records, and each is assigned a 32 byte (3.4E+38) random value. What is the likelihood to guess one of the random values? Context This comes up in information security context: ...
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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
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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
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1answer
34 views

Probability that a random variable is among the top k out of n when ordered

Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$. What is the probability distribution that any given $X_i$ is among the ...
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1answer
38 views

$E(Y_i|X_i = 1)$ where $Y_i = X_i + U_i$ with $X_i$ being Bernoulli and $U_i$ being Normal

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
1
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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
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1answer
48 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
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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 ...
0
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0answers
33 views

Quantile as solution to minimization problem

I'm studying basics of quantile regression now and I have trouble prooving that $\tau-$th quantile of real-valued random variable $Y$ is a solution to the following minimization problem (in the ...