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

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2answers
28 views

Find the expected frequency of some state in a state sequence of length N given a transition matrix M

I can represent stochastically-articulated sequences of states using a transition matrix M where a given entry in cell (i,j) corresponds to the probability of state j given that the current (or, most ...
-2
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5answers
48 views

PDF of function of uniform random variable [closed]

Why PDF of $g(X)=X^3$ is not uniformly distributed, when X is uniform random variable between $(0,1)$? As for every value of X there is unique value of $g(X)$, hence the probability density of $g(X)$ ...
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1answer
51 views

Existence of a global maximum of a function defined with the moment-generating function

Can someone give me an idea how to prove the following exercise? Let $Z$ be a real-valued random variable whose moment-generating function $m_Z$, with $m_Z(\gamma)= E\left[ \exp(\gamma Z) \right]$, ...
2
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0answers
59 views

If $x_n\in S$ and $x_n\to x$ then $x\in S$

Suppose $S$ is the support of a univariate cdf $F$. If $x_n\in S$ is a sequence of reals such that $x_n\to x$ then show that $x\in S$. I believe the question is actually very very simple, in the ...
3
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1answer
70 views

When do we say independence is the probability of intersection being equal to the product of probabilities? [duplicate]

This is something I never really got in either Elementary Probability Theory or Advanced Probability Theory because my professors mainly discussed independence between 2 objects. Please tell me if my ...
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0answers
14 views

What are some error measures used for fitting PMFs?

I have a given PMF, $f_X(x)$, and am trying to create a fitted PMF, $g_X(x)$, that comes "as close as possible" to it, but am not sure what to use as a measure of fit. Simply minimizing standard error ...
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1answer
17 views

How to prove **$k_1$-wise independence implies $k_2$-wise independence if $k_1 \geq k_2$**

The Definition 1 shows the the meaning of k-wise independence. So can anybody help to prove $k_1$-wise independence implies $k_2$-wise independence if $k_1 \geq k_2$? By the way, I did not ...
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0answers
14 views

Sub gaussian concentration for Lipschitz functions

It is well know that: if $f:\mathbb{R}^m\to\mathbb{R}$ is a Lipschitz function with Lipschitz constant $L$, and $X_1,\dots X_m$ are i.i.d random variables s.t. $X_i\sim N(0,1)$, then for any $t>0$ ...
1
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0answers
33 views

$E(x\mid xy)$ linear in $xy$

I read a paper that makes an assumption that $E(x\mid xy)$ linear in $xy$ where $x$ and $y$ are independent. I was wondering what is fundamental about this assumption, or in other words, what ...
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1answer
21 views

Equivalence of the second moment of two random variables when their first moments and covariance with a third random variable are equal

I'm trying to check under which conditions the standard deviation of two random variables is identical when I know some properties about other moments of these random variables. I suppose that their ...
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1answer
34 views

Equivalent definitions for expected value of random variable

Let $(\Omega, P)$ be a probability space. One definition for the expected value of a random variable $X$ is $$E(X)=\sum_{x\in \mathbb{R}} xP(X=x).$$ The notes I am reading say that this definition ...
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1answer
26 views

Using Gronwall's Inequality with Random Variables

Currently, I've been working with an SDE and trying to get a bound on moments. I have it down to something of the following form: $$X(t)^p \leq a(t) + \int_0^t X(s)^pY(s) ds + \int_0^t X(s)^p dW_s$$ ...
1
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1answer
67 views

Proof that random variable is almost surely constant

If a random variable $X : \Omega \to \mathbb R$ is $\{ \emptyset, \Omega \}$-measurable, then it is constant. I want to generalise this result: Now if $\mathcal G$ is a $\sigma$-algebra such that ...
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0answers
41 views

GARCH model, expectation of volatility?

Consider a time series $\{r_t\}$ following a standard GARCH(1,1) model, i.e., $$ r_t = \sigma_t \epsilon_t,$$ where $\epsilon_t \sim N(0,1)$ and are i.i.d, and $$\sigma_t^2 = \omega + \alpha_1 ...
1
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1answer
23 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
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0answers
29 views

Number of same degree vertex pairs between two random graphs

I am considering the random graphs generated by the Erdős-Rényi model for this question. Random Graphs as Models of Networks by Newman is a reference on this topic. A random graph $\Gamma_{n,p}$ has ...
1
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1answer
41 views

Are random variables in a tail σ-algebra in the same probability space?

Let $X_1, X_2, ...$ be random variables. Define $\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, ...)$ and $\mathscr{T} = \bigcap_{n} \mathscr{T}_n$, the tail σ-algebra of $X_1, X_2, ...$. When defining a ...
2
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1answer
62 views

Are the running products of iid RVs independent?

Are the running products of iid RVs independent? Let $Y_0, Y_1, ...$ be independent random variables with $P(Y_n = 1) = P(Y_n = -1) = 1/2 \ \forall n = 0, 1, 2, ...$ (*) Define $X_n = Y_0 Y_1 ... ...
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0answers
22 views

How can I get the pdf of $\cos\theta_1-\cos\theta_2$ when $\theta_1$ and $\theta_2$ are i.i.d. uniformly distributed?

How can I get the pdf of $X=\cos\theta_1-\cos\theta_2$ when $\theta_1$ and $\theta_2$ are i.i.d. uniformly distributed over $[-\pi,\pi]$? I have got the expression of the pdf of ...
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1answer
37 views

Random variable with all higher order moments zero?

Is there a random variable with finite first and second moment but all higher order (non-central) moments are zero?
2
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1answer
37 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
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1answer
54 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
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2answers
69 views

Distribution of Summation of two discrete random variables

Here, $X$ and $Y$ are two non-negative independent discrete integer-valued random variable and the support set of $k_1$ and $k_2$ are $ \{ 2,3,...,7 \}$ and $ \{ 5,6,...,12 \} $ respectively. We ...
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2answers
38 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
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1answer
49 views

Continuous mapping theorem - counterexample

The continuous mapping theorem states that Let $g: R^n \rightarrow R^k $ be continuous in every point of a set $C$ such that $\mathbb P\left(X\in C\right)=1$. If $X_n \xrightarrow{d} X $ then ...
1
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1answer
37 views

Determining Probability Generating Function from Probability Mass Function and Convergence

I am trying to solve the following: Suppose $X_{nk}, k=1,2,\ldots,n, n≥ 2$ are i.i.d. random variables $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
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2answers
237 views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
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1answer
35 views

Application Problem: Expectation and Variance of Compound Poisson Process

I am solving the following: Let $Y1, Y2,…$ be a random sample from $\Gamma(p,a)$ distribution, where p and a are positive real numbers. $Y$ is damage in thousands of dollars caused to a car in an ...
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0answers
22 views

Is the product of two sub-Gaussian random variables a sub-Gaussian random variable?

If not, is there any way to make it hold? Note: the random variable $x$ is called $σ^2$-sub-Gaussian if $E[e^{tx}]≤e^{t^2σ^2/2}$.
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1answer
33 views

Power Spectral Density Approximation

Let $X_t$ be a zero-mean, stationary random process. Let $X_f$ be the Fourier transform of $X_t$; $X_f$ is also a random process, but as a function of $f$. Let us denote the power spectral density ...
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0answers
52 views

Application of Slutsky's Theorem to the Convergence of Sum of R.V.

Let $X_1, X_2,…, X_n$ be i.i.d. $U(−\theta,\theta)$. Show that $Z_n \to N(0,\sqrt{5/9})$ in distribution, where $Z_n ...
4
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2answers
181 views

Impact of random numbers on the eigen-values

How do the eigen-values of the following tridiagonal matrix ($A$) change when adding random numbers $R_i$ (with a normal distribution with the mean 0 and variance $m$) to its diagonal. A is a square ...
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2answers
31 views

Comparison between two exponentail random variables

A and B are exponentially distributed with parameter $\alpha$ and $\beta$. A and B race with each other continuously. $N_b$ denotes the number of times B wins before A wins single time. Find $P (N_b ...
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2answers
54 views

Showing that infinite product of random variables goes to zero: $\prod^\infty X_i \rightarrow 0 \text{ a.s.}$

I am doing the following exercise: Let $X$ be a strictly positive rv with $\mathbb E[X]=1$ but $X \neq 1$ almost surely. Let $X_1, X_2 \dots$ be iid with same distribution as $X$. Now let $M_0=1$ and ...
1
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1answer
28 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
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1answer
30 views

Characteristic Function and Convergence in Distribution of Sequence of R.V.

I am trying to solve the following: Let $X_1,X_2,...$ be a sequence of random variables with $P(X_n=\frac{k}{n})=\frac{1}{n}, k=0,1,2,...,n$. Find the characteristic function of $X_n$ and show that ...
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1answer
24 views

Working with the random variable $\log X$ instead of $X$

Suppose I have a positive stochastic process $X_t$. I'd like to compute certain properties about $X_t$, but suppose I can't and instead I can compute properties about $\log(X_t)$. Can I say anything ...
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0answers
19 views

How to deal with set-valued(set in $\Re^n$) random variables?

I'm trying to attack a problem where the random variable are sets i.e set-valued random variable. Suppose $S = \{X_1, X_2,\cdots,X_n\}$ is a set of sets($X_i$) and $f(X_i)$ is the probability ...
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1answer
40 views

Deriving Probability Density Function from Probability Generating Function for Random Sum

I am trying to solve the following: Let $X_{i}$~$Geometric(q) i=1,2,...,N$ with $q=1-p, 0<p<1$. $N$~$Geometric(p)$. Define $Y=\sum_{i=1}^{N}X_i$ and assume each $X_i$ is i.i.d. and ...
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1answer
45 views

$E(f(|X_n|))$ property implies uniform integrability?

This is exercise 6.10 in Resnick's book "A Probability Path". We're given a sequence of random variables $(X_n)$ and an increasing function $f: [0, \infty) \rightarrow [0, \infty)$ such that $$ ...
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1answer
44 views

Showing Convergence in Distribution for Conditional Random Variable

I am trying to prove the following: Let $X$ and $Y$ be random variables such that $Y | X = x$ ~ $N(0, x)$ with $X$ ~ $Po(\lambda$). Show that $\frac{Y}{\sqrt{\lambda}} \to N(0,1)$ in distribution as ...
3
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1answer
42 views

Computing Conditional Characteristic Function

I am trying to compute the characteristic function of the following: Let $X$ and $Y$ be random variables such that $Y\mid X = x\sim N(0, x)$ with $X\sim\mathrm{Po}(\lambda)$. Find the characteristic ...
2
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1answer
43 views

How can I find the density of $E[X\mid Y]$ when $(X,Y)$ is gaussian

I was tying to prove the following: Given $(X,Y)$ a centered gaussian vector in $\mathbb{R}^2$ with the following covariance matrix $$ \Sigma = \begin{bmatrix} \sigma^2_x & \sigma_{x,y} \\ ...
2
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1answer
72 views

Showing Convergence in Distribution of Continuous Function of Sums of R.V.s

I am trying to solve the following: Let $X_1, X_2, . . .$ be i.i.d. r.v.s with mean $\mu$ and positive, finite variance $\sigma^2$, and set $Sn = \sum_{k=1}^{n} X_k, n ≥ 1$. Suppose that $g$ is twice ...
3
votes
1answer
57 views

Proving a Variation of the the Central Limit Theorem

I am trying to prove the following: Let $X1, X2, . . .$ be positive, i.i.d. r.v.s with mean $\mu$ and finite variance $\sigma^2$, and let $S_n = \sum_{k=1}^{n} X_k$ , $n \ge 1$. Show that $\frac{S_n ...
2
votes
1answer
39 views

How do you proof that F is a distribution function, when x > 0

I hope that someone could help me solve this question of my textbook: Let F (x) = e^(−1/x) for x > 0 and F (x) = 0 for x ≤ 0. Is F a distribution function? If so, find its density function. How do ...
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1answer
37 views

Convergence in Probability for a Sequence of Random Variables

I am trying to solve the following: Let $\{X_n, n ≥ 1\}$ be a sequence of i.i.d. random variables with density $f(x) = e^{−(x−a)}$, for $x ≥ a$ and $f(x) =0$, for $x < a$. Set $Y_n = \min(X_1, ...
2
votes
1answer
29 views

Computing Distribution of Conditional Expectation of Gaussian RV

I am trying to compute distribution of the following random variable \begin{align*} E[(X-E[X|Y])^2|Y] \end{align*} where $X \sim \mathcal{N}(0,\sigma^2_x)$ and $Z \sim \mathcal{N}(0,\sigma^2_Z)$ where ...
1
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0answers
47 views

What is the product of two independent random variables (as mentioned below)?

Let $X$ and $Y$ be two random variables with: $\begin{equation} f_{X}(x) = \begin{cases} e^{-\lambda T} & \text{if } x = 0;\\ \lambda T e^{-\lambda T(1-x)} & \text{if } 0 < x \leq ...
0
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0answers
17 views

Is this expression true for moments of random variables?

Suppose $X_1(t), \cdots, X_n(t)$ are random variables of a continuous time stochastic process. Suppose for any $p>1$, $\sup_{t \geq 0} E\left[\sum_{i=1}^n X_i(t)^p \right] < K_p$ where $K_p$ is ...