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

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1answer
52 views

Show that if two random variables sequences are pairwise independent then the limits are independent, too.

Two sequences $X_1, X_2, \ldots, Y_1, Y_2,\ldots : (\Omega, \mathcal{F},\mathbb{P}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ of real random variables such that $\forall n \ X_n, Y_n $ are ...
2
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2answers
44 views

Independent, random variables with equal distribution satisfy: $\lim_{n \to \infty}\mathbb{P}\left(X_{n+1} > \sum_{i = 1}^{n}X_i\right) = 0$

$X_1, X_2, \ldots$ are independent, non-negative, real random variables with equal probability distribution. Show that $$\lim_{n \to \infty}\mathbb{P}\left(X_{n+1} > \sum_{i = 1}^{n}X_i\right) ...
0
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0answers
15 views

How to use symmetry of transition rate matrix in a continuous-time Markov chain?

This is part of a bigger question, so I have to change the question a bit to focus on the point. We have a continuous- time Markov chain with the following transition rate matrix: $$Q= \begin{pmatrix} ...
1
vote
1answer
22 views

Binomial distribution problem

This is a question in my statistics book: I could answer perfectly till part (c) but honestly, I have no idea why the answer to (d) is $0.4005$. Why is the top divided by $0.065^2$ and where did ...
0
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2answers
69 views

Entropy of sum of two Uniform random variables

say $X$ and $Y$ are two identical, independent and discrete Uniform random variables and $Z=X+Y$. I do not know more about the random variables. Assuming $H(\cdot)$ to be the entropy of a random ...
1
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1answer
44 views

Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
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3answers
52 views

2 independent poisson random variables probabilities and 2 different proofs

So, in the above exercise I was wondering if I could get some help with : 2.1 - I was told moment generating functions could help me prove that, but I can't get it 2.2 - I don't get how to start ...
1
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2answers
45 views

How do I transform an r.v. using the floor function? (exponential distribution)

Just had a bash at this question for my Intro to Maths Stats module...I got to the end with a probability density function rather than a probability mass function, namely $f_Y(y) = \lambda a ...
1
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1answer
19 views

Mean and STD of a max/min of an exponentially distrubuted iid random variable

Let $S_1, S_2, S_3, ...$ be a sequence of independent, identically distributed (iid) random veriables, each exponentially distributed with a mean of $\mu_S$ (hence $\sigma_S = \mu_S$). Let $M_n = ...
2
votes
1answer
36 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
3
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2answers
53 views

How to show a random variable diverges from its mean?

I am trying to prove the following statement which seem not to be very complicated but I cannot find a straightforward way to prove it (can it even be wrong?): Suppose $X_n$ are a sequence of random ...
6
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0answers
174 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
2
votes
0answers
73 views

A measure has no point masses: is it absolutely continuous?

I have a question about measure theory. Let $\mu$ be a measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R})$. Assume that $\mu$ has no point masses - i.e. for every $a \in \mathbb{R}$, $\mu({a})=0$. Can ...
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4answers
53 views

Number of Rolls of Fair Dice to get '6' and '5'

A Fair Dice is Thrown Repeatedly. Let $X$ be number of Throws required to get a '$6$' and $Y$ be number of throws required to get a '$5$'. Find $$E(X|Y=5)$$
1
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1answer
31 views

problem involving a bivariate gaussian

I need some help with this exercise, I tried to do this but my calculations seem to go nowhere, any help or hint can be very useful
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0answers
22 views

using Cochran's theorem for sample variance where samples are not identical

IS is possible to use Cochran's theorem to prove that the sample variance of normal variables is chi-square in the case the variables are independent but not identical - they all have the same ...
1
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0answers
44 views

Second moment of random variable in the integral form

Let $X_1,\dots,X_n$ are i.i.d. samples from uniform distribution on $(0,1)$. Let $\hat F_n$ be their modified empirical distribution function defined by $$ \hat ...
0
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2answers
37 views

What is $\operatorname{Var}[aX+bY+c]$?

I know that $\operatorname{Var}[aX+bY]=\operatorname{Cov}[aX+bY,aX+bY]=a^2\operatorname{Var}[X]+2ab\operatorname{Cov}[X,Y]+b^2\operatorname{Var}[Y]$ (by expanding $(ax+by)(ax+by)$ and letting ...
0
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1answer
43 views

Covariance of random variables which don't have variances.

Whether there is a covariance of two random variables, which both don't have variances? Is existence of the variances of two random variables implies existence of covariance? Thanks in advance.
1
vote
1answer
35 views

Product of sub Gaussian RVs

Suppose that $X,Y$ are two independent sub-Gaussian RVs. Let $Z=XY$. Is $Z$ also sub-Gaussian? Can someone provide any reference presenting some basic properties of sub-Gaussian RVs. Thank you in ...
0
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1answer
39 views

Multivariate normal distribution independet iff uncorrelated

I found a few threads about this but none of them answered my question. I am supposed to show that if you have random variables $X_1$,$X_2$ that are gaussian distributed and they fulfill that ...
1
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1answer
25 views

Finding efficiency of an estimator for Poisson random variables

$\newcommand{\eff}{\operatorname{eff}}$ I am asked to derive the efficiency of the estimator $\hat{\lambda}_1 = \frac{1}{2}(Y_1+Y_2)$ relative to $\hat{\lambda}_2=\bar{Y}$, where $Y_1,Y_2,\ldots,Y_n$ ...
0
votes
0answers
30 views

Expected Value With Signum

So I am trying to show convergence of a filter, and in order for it to converge, I need the following condition to hold: $ E\{ \mathbf{s} x sgn(\mathbf{h}^{T} \mathbf{s} x) \} \; \alpha \; E \{ ...
0
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0answers
25 views

Finding the PDF of an nonlinear transformation

I'm trying to estimate the gain in a frequency $\omega$ of an ARX model as below: $$y[k]=\sum_{i=1}^n a_i y[k-i]+\sum_{i=0}^m b_i u[k-i]$$ Where $u[k]$ is the input of my system. I have already ...
0
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0answers
8 views

Solution verification on standard normal random variable.

The problem I am solving is as follows: A certain IQ examination has a score that is normally distributed with mean $100$ and standard deviation $14.2$. What is the probability that a randomly chosen ...
1
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1answer
43 views

from one dimensional Gaussian to two dimensional Gaussian

Given $x$ is a random variable, which is one-dimensional Gaussian distributed, such that $$ x \sim \mathcal N(\mu, \sigma^2).$$ Given $\theta$ is some constant, what would be the density function for ...
3
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4answers
143 views

Does this sum converge or diverge?

Does the infinite sum $\large{\sum_{n=1}^\infty \frac{1}{n^{x_{\small{n}}}}}$ converge if $x_n$ is a random variable (generated within each term) that takes values between $0$ and $2$ with equal ...
0
votes
1answer
27 views

Dealing with this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. $f_1>0$ is a positive constant and $f_2,f_3,f_4\ldots$ are positive functions of one variable. ...
1
vote
1answer
52 views

$n$-dimensional Gaussian distribution: Iso-density manifold. What else?

Let X be a random variable that follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and covariance matrix the $n\times n$ symmetric positive matrix $\Sigma$, i.e. ...
0
votes
1answer
21 views

Question regarding Poisson random variable

I'm reviewing a question I did for the Poisson random variable and I can't remember how I got the the answer to part b). The problem goes as follows: "Suppose that the number of typographical erros on ...
0
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0answers
40 views

If $A=\{R_1\le\frac{1}{2}\}$ and $B=\{R_2\le\frac{1}{2}\}$, find $P(A\cup B)$

Let $(R_1, R_2)$ have the following density function $f_{12}(x,y)=\cases{ 4xy & \text{if } 0\le x,y\le1, x\ge y\cr 6x^2 & \text{if } 0\le x,y\le1, y>x }$ If ...
1
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2answers
70 views

Is expectation countably additive?

I am supposed to prove that Let $X \ge 0$ be a random variable defined on $(\Omega, \mathcal{A}, > P)$ and $\mathbb{E}[X] = 1$. Define $Q: \mathcal{A} \to \mathbb{R}$ by $Q(A) = ...
1
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1answer
21 views

Mean and STD of a sequence based on Exponential Random Variable

Say I have a sequence $S$ that is a exponentially random variable with mean $\mu$. Now say I create another other sequences from this: $T$ which is $T(n) = 2(S(n+1)-S(n))$. I know that theoretically, ...
0
votes
1answer
29 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in ...
1
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2answers
47 views

Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = ...
1
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0answers
20 views

Convergence of sequences of random variables

Let $X_1, X_2, ...$ and $Y_1, Y_2, ...$ be two sequences of nonnegative random variables. Assume that each $n$ random variable $Y_n$ is uniform in the interval $[0, X_n]$. Show that if ...
2
votes
1answer
42 views

P.d.f. of $XY$, where $X, Y$ are independent uniformly distributed over $[0,1]$ [duplicate]

I tried to change the variables: Let $U=XY$ and $V=Y$; so then the Jacobian is $1/v$. So joint pdf $g(u,v) = f(x,y)\cdot (1/v) = 1/v$ Would you then integrate over $v$ from $0$ to $1$ to get the ...
1
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0answers
123 views

Limit of sequence of integral related i.i.d. observations

Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat ...
1
vote
2answers
37 views

Expected value of a the reciprocal of a random number

If I selected a real number at random from the interval (0.0,1.0), assuming a uniform distribution, the "expected value" would be 0.5. (I am not certain I am using the phrase correctly; I mean, if I ...
2
votes
1answer
19 views

Discrete distribution with the minimum variance

Consider a discrete random variable $X \in \{x_1, x_2, \ldots, x_n\}$, where $n < +\infty$ and $x_1 < x_2 < \ldots < x_n$. Let pose $p_i = \text{Pr}(X = x_i)$, with $\sum_{i=1}^N p_i = ...
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2answers
46 views

What's wrong with this random variable proof?

Let $X$ be a Binomial random variable $\sim B(p, n)$. Show that for $\lambda > 0$ and $\epsilon > 0$, $P(X - np > n\epsilon) \le \mathbb{E}\{\displaystyle e^{\lambda(X - np - ...
4
votes
2answers
71 views

If the sum of two independent random variables is in $L^2$, is it true that both of them are in $L^1$?

Let $X$ and $Y$ be two independent random variables. If $\mathbb E(X+Y)^2 < \infty$, do we have $\mathbb E |X| < \infty$ and $\mathbb E |Y| < \infty$? What I actually want is that $X$ and ...
0
votes
1answer
40 views

One problem on random vectors

$e_i$'s are $n$ - dimensional random vectors and any two different random vectors are uncorelated . I need to prove $$E[||\sum_{k=0}^{\infty} a_k e_k||^2] = E[\sum_{k=0}^{\infty} a_k^2 ||e_k||^2]$$
0
votes
1answer
30 views

Running time and standard deviation of Marsaglia and Bray's Polar Method

I have some trouble calculating the standard Deviation and the mean running time of the polar method by Marsaglia and Bray. More specifically the repeat portion of the algorithm. In this case the ...
1
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1answer
25 views

Expectation of the square of the minimum of iid positive random variables

Let $X_1, X_2$ be i.i.d., positive random variables with $E[X_i] < \infty$ (but $E[X_i^2]$ might be $\infty$). $Y := \min \lbrace X_1, X_2 \rbrace$. I want to show that $E[Y^2] < \infty$. The ...
2
votes
2answers
43 views

Convergence of random variables in metric spaces

Let $S$ be a metric space equipped with a distance function $d$, and let $X_n,Y_n$ be sequences of random variables having values from $S$. Suppose that $X_n$ converges in distribution to some random ...
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0answers
27 views

Probability of divergence of a sum of random variables with constant positive expectation

I've encountered the following question: suppose $X_n$ is a sequence of positive random variables such that $\mathbb{E}(X_n)=1$ for all $n$. Does it follow that $\sum X_n$ diverges almost surely? ...
0
votes
1answer
31 views

Cauchy random values in a interval [a, b]

How do I generate random numbers following a Cauchy distribution in a given interval [a, b]. I tried using explained here Trucated distribution, but did not succeed
2
votes
1answer
56 views

Weak convergence: equivalence of definitions

Consider a sequence of random variables $(X_n)_{n\geq 0}$ and a random variable $X$. How to prove that the two following definitions of weak convergence are equivalent? Def 1 $(X_n)_{n\geq 0} ...
-1
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1answer
54 views

Generate exponential random values in a given range [duplicate]

Need to generate random values ​​that follow an exponential distribution on an interval [a, b​​]. I tried using explained here Trucated distribution, but did not succeed