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

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38 views

Deriving the autocorrelation function for the ARMA model

Definitions The ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. Question To derive the ...
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1answer
37 views

Does the probability commute with limit?

Does the probability commute with limit? For example, is it true that for $(N_t)_t$ random variables which take values in the set of natural numbers, $$ \mathbb{P}(\underset{t \rightarrow 0}{lim} ...
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1answer
39 views

Yule walker equation limited matrix size

Definitions For an ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. It is straightforward to show that ...
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2answers
60 views

How to determine if a random variable is $\mathcal F$-measurable?

For example : Consider the state space $\Omega = \mathbb{R}$, the $\sigma$-algebra, $\mathcal{F} = \{(-\infty, 0], (0, \infty), 0, \mathbb{R}\}$ and the random variable $X : \Omega \rightarrow ...
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3answers
29 views

Discrete random variable. Tossing a coin.

Two coins are simultaneously tossed until one of them comes up a head and the other a tail. The first coin comes up a head with probability $p$ and the second with probability $q$. All tosses are ...
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1answer
35 views

Probability to iteratively and independently remove $n$ elements until all gone

The problem is as follows: Let S be a set of n elements. At the first stage each element in S is in- dependently removed with probability p. Those elements not removed constitute the set S1. ...
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1answer
29 views

A question involving independent random variables and indicator random variables [closed]

Let $X$ and $Y$ be two independent random variables. It is true that $1_{X > 0}$ and $1_{Y > 0}$ are independent? Why yes / not? Thank you!
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1answer
19 views

compute conditional expectation with respect to sigma algebra

I'm studying stochastic and I'm stuck at the following problem: I ask myself how to compute this conditional expectation: Let $X$, $Y$ be two independent random variables in $L^1$. What is ...
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0answers
27 views

Independence of random variable through bitwise rotation

Say, I have a property that requires two $w$-bit random variables $X_1$ and $X_2$ to be independent and uniformly distributed when I perform addition modulo $2^w$. Now, instead of taking two ...
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1answer
32 views

When can I switch sup and functions

I have a sequence of random variables $X_k$. Under which conditions can I say that $$f\left(\sup_{j \le k} X_k\right) = \sup_{j \le k} f(X_k)$$? Would having $\limsup X_k = \lim_{k \to \infty} ...
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1answer
78 views

$E[x_i^2 x_j^2]$ for white Gaussian noise

If $x_n$ is a discrete time random signal and is white Gaussian noise (ergodic and WSS) so $$E[x_n x_{n+l}]=\sigma ^2 \delta (l)$$ and $$E[x_n]=0$$ Where $n \in \mathbb{R}$ and $l\in\mathbb{R}$ ...
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0answers
32 views

max eigenvalues distribution of tridiagonal symmetric random matrix with zero on the main diagonal

Can any one can refer me to a paper discussing how to find the max eigenvalue distribution for the following matrix where all the $\lambda$'s are random variables \begin{align} \small %%%%% ...
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0answers
26 views

Conditional Probability and mutual independence

i have a question. On a Probabilityspace we have RVs $X_i:\Omega\mapsto\{x, y, z\}$ and $Y_i^g:\Omega\mapsto\{-1, 1\}$, with $g\in\{x, y, z\}$ and $i\in\{1,2\}$. We know, that ...
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1answer
18 views

random variables independence

I need to check if Z and W are dependent or not. X,Y ~ Exp(2) Then I define: Z=X-Y , ...
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0answers
8 views

Probability density function of $|s^{\dagger} M z |^2$ where $z \sim \mathcal{CN}(\mu,R)$

What could be done to find the probability density function of $$|s^{\dagger} M z |^2$$ where $z \in \mathbb{C}^{N \times 1} $ is the proper complex Gaussian vector i-e $z \sim \mathcal{CN}(\mu,R)$, ...
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1answer
19 views

Generalization of Lebesgue Convergence theorem

A generalization of Lebesgue Dominated Convergence Theorem tells us that if $f_n \to f$ in probability and $|f_n| \le g \in L^p$, then $|f| \in L^p$ and $f_n \to f$ in $L^p$ norm. Suppose $f_n \to f$ ...
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1answer
23 views

Finding the moments of normal variables

How to calculate the moments of a normal random variable with mean $\mu$ and variance $\sigma^2$? Using integration by parts we get the recurrence relation (calling $a_n = E(X^n)$) $$\begin{cases} ...
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2answers
129 views

Divergence of random series

So, suppose that $X_n$ is a sequence of independent identically distributed random variables with Bernoulli distribution with parameter $p$. Now, consider the random series $$\sum_{n=1}^{\infty} ...
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2answers
23 views

Given 3 white balls and 1 black ball in a box.Take out ball by ball until getting a black one.Calculate the expected value for number of takes

Given a container with 3 white balls and 1 black ball. We take out balls randomly from the container, one by one, until a black one pop out. Calculate the expected value of number of balls ...
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1answer
36 views

Let $X, Y$ be independent random variables, $X \text{~} Pois(2), Y \text{~} Pois(3)$. calculate: $P(X + Y \le 1)$

Another previous year exam question: Let $X, Y$ be independent random variables, $X \text{~} Pois(2), Y \text{~} Pois(3)$. calculate: $P(X + Y \le 1)$ What is the right way to approach such ...
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1answer
43 views

Can $Z$ be a function of $X$ and $Y$?

The three random variables $X,Y$ and $Z$ are pairwise independent. Can $Z$ be a function of $X$ and $Y$, and by this I mean $Z=f(X,Y)$, where $f$ is a function?
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0answers
12 views

Exact formulation of the law of large numbers for i.i.d. Paretian variables

If $X_1, X_2, ...$ are i.i.d. Pareto-distributed variables with parameter $\alpha$, I have read (in a heuristic source) that $\frac{1}{n^{1/\alpha}} \sum_{k = 1}^n X_k$ fulfills some law of large ...
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1answer
65 views

Radius of convergence of power series (random variables)

Let $z_n$ be a sequence of independent identically distributed random variables and let $$f(x)=\sum_{n=0}^\infty z_nx^n$$ be a random power series. How to show that the radius of convergence is ...
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2answers
40 views

Is E(X|Y) $\sigma(X)$-measurable?

From the definition of conditional expectation, we have that E(X|Y) is $\sigma(Y)$-measurable. I wonder if E(X|Y) is also $\sigma(X)$ measurable? It seems to be true since E(X|Y) is a "coarser" ...
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0answers
15 views

Characteristic function of an independent variable, does it involve complex values?

Let $$ x_k = \begin{cases} 1 & \mathrm{prob} (1/2)\\ -1 &\mathrm{prob} (1/2) \end{cases}$$ be independient random variables. Show that the characteristic function of the random variable ...
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1answer
21 views

If r.v $X$ is independent of a vector of r.vs, will $X$ be independent of any linear combination?

Let $X$ be a random variable, and $\mathbf{Y}=<Y_1,Y_2,...,Y_n>$ be a vector of random variables. If $X$ is independent of $Y_i \forall i=1,2,3,...,n $, will this $X$ be independent of any ...
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0answers
23 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
2
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1answer
55 views

The expectation for large numbers random sequence

Suppose $$H=h_1^2+h_2^2...h_N^2$$ where $h_i$ is i.i.d Gaussian distributed random variable with zero mean and unit variance, i.e., $h_i \sim N(0,1)$. By the strong law of large numbers, we have: ...
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1answer
106 views

Proving a Poincare inequality using Stein's characterization

Question: Let X be a standard Gaussian r.v. Use Stein's characterization $Ef'(X) = E(Xf(X))$ to prove the Poincare inequality $E|f(X)-Ef(X)|^2 \leq E|f'(X)|^2$. This looks like Markov's ...
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1answer
11 views

Conditional cdf of exponential variable

Given the rate parameter for an exp r.v, I am able to calculate conditional pdf and mean on the condition of A > c. For conditional pdf I calculate P(A>c) and divide that by the pdf of the r.v. I ...
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1answer
37 views

How to understand the proof of E[X+Y] = E[X] + E[Y]?

I am studying probability and trying to follow an example in my textbook discussing expectation of the sum of random variables. But I'm having trouble following it. $E[X+Y] = ...
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2answers
23 views

Does orthogonality between two linear combination of random variables imply indepedence?

Assume we have a vector of random variables $U=\langle u_1,u_2,u_3,....,u_n \rangle$. random variables of $U$ are not necessary independent random variables. Now I have two vectors of coefficients: ...
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0answers
15 views

product of independent random variables

Let $X \sim U(-10,10)$ and $Y$ have a pdf $f_Y(y) = \dfrac{1}{t} y ^ {(1/t) -1} $, $0\leq y\leq 1$. If $X$ and $Y$ are both independent, what is the pdf of $XY$? Here is my attempt: Let $W=XY$, ...
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3answers
58 views

Distribution of time that a flashlight can operate

The lifetimes of batteries are independent exponential random variables , each having parameter $\lambda$. A flashlight needs two batteries to work. If one has a flashlight and a stockpile of n ...
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1answer
34 views

Linear transformation of a random variable

Is there a name for the class of distributions where a linear transformation does not alter the underlying distribution? In particular, adding a constant changes the mean and multiplication by a ...
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1answer
20 views

Linear combination of a sequence converging in distribution (weakly)

I've been trying to prove that if $X_n \rightarrow X$ in distribution, that is for $F_n, F$ - distribution functions of $X_n, \ X$ resp we have: $$\forall x: \ F \ \text{is continuous in} \ x : \ ...
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1answer
15 views

Convergence of ratio of linear combination of iid random variables to their sum

Suppose $x_1,x_2,...x_n$ are i.i.d. normal random variables, $a_1,a_2,...a_n$ are some positive constants. Could we have following equation $$\sum_{i=1}^n\frac{a_ix_i^2}{\sum_{i=1}^n ...
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0answers
32 views

Finding the expected value of a A Gaussian voltage distributions?

A Gaussian voltage random variable $X$ has a mean of $ \over X $ = 0, and variance of $9$. The voltage $X$ is applied to a square-law, full-wave diode detector with a transfer characteristic $Y = ...
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1answer
39 views

How to find distribution of order statistic

Let $X_i$ be iid random variables with common density $f$ and distribution $F$. Let $Y_k = X_{(k)}$ be the k-th order statistics (that is, $Y_1 = X_{(1)} = \min(X_i)$ etc.). Show that the joint ...
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1answer
40 views

Joint distributions

Let $X, Y$ be continuos random variables of densities $f_X, f_Y$. Let $Z = \begin{pmatrix} X \\ Y \end{pmatrix}$. When is $Z$ continuos? And in this case, how to express its density with respect to ...
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34 views

conditional probability on a triangle

Consider a triangle with vertices $(0,0), (0,1),(1/2, \sqrt{3})$. Suppose that the random vector $(X,Y)$ is uniformly distributed on this triangle. Compute $f_{Y}(y)$ and the conditional probability ...
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3answers
45 views

Sum of two independent binomial variables

How can I formally prove that the sum of two independent binomial variables X and Y with same parameter p is also a binomial ?
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1answer
51 views

What is the joint distribution of two random variables?

Today I was thinking about this and I have the feeling I am missing something obvious, but I can't seem to solve it. Suppose we have a continuos random variable $X$ with density $f_X(x)$. Let $Y = ...
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1answer
18 views

Discrete fourier transform of random noise interpretation

Let's say i have a real valued random noise $\eta(t)$, for which I took $N$ samples with $N$ even, i have therefore a vector of sampled values. I want to compute the Discrete Fourier transform of the ...
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4answers
43 views

Can anyone give me a hint how to start this problem?

Let $X$ be a random variable such that $P(X \leq 0)=0$ and let $\mu =E(X)$ exist. Show that $P(X \geq 2\mu) ≤ \frac{1}{2}$. I don't really know how to start this one... so any hint will be ...
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2answers
71 views

If $X$ is a continuous random variable and $Y$ is a discrete random variable, is $P(X=Y) = 0$?

I have a general question. Lets say I have $2$ random variables $X$ - Continuous Random Variable $Y$ - Discrete Random Variable For all $X$,$Y$ is $P(X=Y) = 0$ ?
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1answer
39 views

Expected Waiting Time

Interesting question our professor raised at lecture today, trying to figure out how to solve it. Goes as follows: "An astronomer goes out at night taking pictures of clusters of comets. Every 5 ...
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30 views

E(x) and Var(x) of a function with two random variables

The problem is "a horizontal force $X$ acts on a surface inclined at an angle $a$ from the horizontal. The force $X$ can be resolved into normal $N$ and tangential $T$ components such that ...
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0answers
15 views

Maximum of Binomial Random Variables

Let $X \sim Bin(n,p)$. Let $\{X_i\}$ be $n$ iid copies of $X$. Let $Z = \max{X_i}$. I want to put upper bounds on $E[Z]$. The variance of $X$ is $np(1-p)$. Following the analogy from the sub-Gaussian ...
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0answers
18 views

How to sum random variables

Let $Z_t = \psi_t |\lambda Z_{(t-1)} + (1-\lambda)\epsilon_t |$ be a random variable where $\epsilon~N(0,1)$ is a Gaussian distributed number, $Z_0 = z_0$ and $\psi \in [-1,1]$ a random variable, ...