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

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0answers
27 views

Convolution of which distribution will give a uniform distribution?

Suppose there are two IID random variables x1 and x2. What should be the distribution of these random variables so that the distribution of x1-x2 is a uniform distribution?
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1answer
15 views

Question Regarding finding the mean and variance of a MGF function?

This question confused me at the end where it says a normal random variable. A breakdown of the answer would be great The Question states: The MGF for the (general) normal distribution is given by ...
1
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2answers
85 views

Do moments define distributions?

Do moments define distributions? Suppose I have two random variables $X$ and $Y$. If I know $E[X^k] = E[Y^k]$ for every $k \in \mathbb N$, can I say that $X$ and $Y$ have the same distribution?
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1answer
32 views

Fourier transform of a random variable

During my research i'm dealing with a stochastic partial differential equation. The random term appearing in my equation is a tensorial random variable: $\boldsymbol{\sigma}(\boldsymbol{x},t)$ Which ...
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1answer
38 views

Volume and Probability of a region given by a random variable

I am currently reading this paper. It is about nearest neighbors of a query point $X_q\in\mathbb{R}^k$ within a point set $P=\{X_i\mid X_i\in\mathbb{R}^k\}$, where the points have distribution $p(X)$ ...
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1answer
16 views

Limiting Behaviour of Root Mean Square Normal Random Variables - Related to Chi-Squared Distribution

Above is my question. I have done the first part - made hard work of it, albeit, but still, it's done. The next part is where I am stuck. Intuitively, it seems (to me!) like we should have $R_n ...
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3answers
27 views
0
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0answers
37 views

$X$ and $Y$ are independent of $Z$, Will any linear combination of X and Y be independent of Z?

If $X$ and $Y$ are independent of Z, will any combination of $X$ and $Y$ be independent of Z? $aX+bY \perp Z$? Will that independence holds if $X \perp Y$?
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0answers
29 views

Determinability

If $X$ and $Y$ are random variables taking values in measurable spaces $(E,\mathcal{E})$ and $(D,\mathcal{D})$ respectively, then we say that $X$ determines $Y$ if $Y=f\circ X$ for some measurable ...
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0answers
47 views

Relationships between Uniform and Pareto Distributions

If $X$ is uniformly distributed over $(a,b)$ and $Y$ is pareto distributed with parameters $(min,c)$, what is the distribution of Z in the following cases? (a) $Z = X + Y$ (b) $Z = XY$ (c) $Z = ...
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1answer
28 views

Covariance of random variable as a function of distribution of noise

Consider the following stochastic difference equation \begin{equation} x(t+1) = x(t) + \nu(t+1) \end{equation} where, $x(t)\in\mathrm{R}$ be one dimensional and $\nu(t)$ be the disturbance with an ...
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0answers
53 views

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find the joint density.

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find $f_{X,Y}(x,y)$ and the marginals $f_X(x)$ and $F_Y(y)$. My attempt: Since the random vector is uniform it will have ...
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2answers
25 views

Range of a random variable - measure theory

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Sometimes, the convention is used that a random variable is a map from a probabilty space to $\mathbb{R}$, but let's not adopt this. So let $X$ ...
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0answers
38 views

Is there some limit result with conditional expectation?

Let $Y$ be a random variable and $\mathcal F_n$ be an increasing sequence of sigma algebras. Do we have some limiting result for $E[Y \mid \mathcal F_n]$? For example, if $$\mathcal F_\infty = ...
2
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0answers
35 views

If $X\geq0$ is a random variable, show that $\lim\limits_{n\to\infty}\frac1nE\left(\frac{1}{X}I\left\{X>\frac{1}{n}\right\}\right)=0$

If $X\geq0$ is a random variable then show that:$$\lim_{n\to\infty} \frac{1}{n} \cdot E\bigg(\dfrac{1}{X}I\bigg\{X>\dfrac{1}{n}\bigg\}\bigg)=0$$ A hint would be most appreciated. I have ...
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1answer
33 views

Covariance expressions given three or more random variables

Suppose we have three random variables $X,Y,Z$, then is it true that \begin{align} E(XYZ)&=Cov(X,YZ)+E(X)E(YZ)\\ &=Cov(XY,Z)+E(XY)E(Z)\\ \end{align} and generally for higher number of ...
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0answers
32 views

Showing Two Random Variables are Indendent Based on their PDF

Let $X_1,...X_n$ be a random sample from population with pdf $F(x|\theta)=\alpha \theta^{-\alpha}x^{\alpha -1}$ where $ 0 < x < \theta$ Show that $\frac{X_k}{X_{k+1}}$ and ...
2
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2answers
35 views

How to imagine squared random variable

Let's have two independent random variables $X$ and $Y$ and two expressions $$E(X^2 Y^2) \dots E(X)^2 E(Y)^2$$ because they are independent then $$E(X^2)E(Y^2) \dots E(X)^2 E(Y)^2$$ Instead of dots ...
2
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1answer
37 views

Gamma distributions and independent random variables.

Assume that two independent random variables $X$ and $Y$ are Gamma-distributed such that $X \sim \Gamma(a,c)$ and $Y \sim \Gamma(b,c)$ with $a, b, c > 0$. How can we see that two random variables ...
1
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1answer
32 views

Product of normal random variables - bivariate normal?

Im wrong about something here, but Im not sure what. As far as I know the product of two normal distributed variables is not normal distributed. However, if the joint distribution of Y and X is ...
2
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0answers
42 views

Show that following mapping is measurable

Let $\{X_{n}\}_{n\geq 1}$ be a sequence of random variables on a probability space, $(\Omega, \mathcal{A},\mathbb{P})$. Define the following mapping: $X : \Omega \rightarrow \mathbb{R}^{\infty}$ ...
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1answer
33 views

Box-Muller Independence Proof by Change of Variables (Help finding the Inverses)

Let $X_1=\cos(2 \pi U_1)\sqrt{-2 \log(U_2)}$ and $X_2=\sin(2 \pi U_1)\sqrt{-2 \log(U_2)}$ wher $U_1$ and $U_2$ are iid uniform (0,1). Prove that $X_1$ and $X_2$ are independent N(0,1) random ...
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1answer
34 views

product of random variables, assymetric and piecewise

Let $X∼U(−10,10)$ and $Y$ have a distribution that is piecewise defined as follows: $$ f_Y(y) = \left \{ \begin{array}{ccc} \dfrac{2}{\sqrt{16y+9}} & if & -\dfrac{9}{16} < y \leq 0 \\ ...
2
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1answer
55 views

Prove Kolmogorov's zero one law using martingales

I am supposed to provide a martingale proof of Kolmogorov's zero-one law. Hint Let $X_n$ be independent random variables and let $\mathcal C_\infty$ be the corresponding tail $\sigma$-algebra. Let ...
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0answers
22 views

Differentiating w.r.t. the boundary of the expected value

I need to solve for general and cont. diff. pdf $g(x)$ $$\frac{d}{db} \int_0^b xg(x)dx$$ Standard Leibnitz rule would give me $$b g(b) + \int_0^b 0 dx$$ the result makes sense - but I'm not sure ...
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2answers
28 views

Integrating density

Say $g(x)$ is the density of a continuous random variable, and $G(x)$ is its cdf. Im trying to understand some of the integral tricks I keep seeing everywhere. I know that $\int g(x) dx = G(x)$. ...
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1answer
97 views

The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$

We have $n$ sensors $X_i$ which estimate the scalar value $\alpha$ with different relative accuracies $\delta_i \ll 1$: $$ x_i = X_i(\alpha) = \xi_i \cdot \alpha, \ \ \ \xi_i \sim N(1, \delta_i) $$ ...
3
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1answer
68 views

Is it true that:$X>Y$ implies $\mathbb Eg(X)>\mathbb Eg(Y)$ for $g$ is strictly increasing function

Suppose $X$ and $Y$ are two random variable. Let X>Y stochastically and $g$ is an strictly increasing function. Is it true that E(g(X))>E(g(Y)) strictly and how to prove it? By $X>Y$ ...
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0answers
35 views

Prove the expectation value of a function of random variables

Consider a random variable $A$ and suppose we look at it the expectation value of $A^m$. Then we have the expectation value of $A^m$: $$= \sum\limits_{n= 1}^N {a^m_n}p $$ Using ...
2
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1answer
38 views

What's the conditional probability mass function of a Poisson random variable less than t given that it and another Poisson r.v. equal t?

The full question I'm working on: Let $X_1$ and $X_2$ be two independent Poisson random variables with mean parameter $\lambda > 0$. Let $T= T(X_1, X_2) = X_1 + X_2$. [Note: You may use the ...
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0answers
10 views

Finding the PMF and probability of financial advisors having a clean record?

According to an agency, 34% of its financial advisors have a clean record. Assume 10 advisors are randomly selected. What is the probability at-least 2 have a clean record? I want to find the ...
3
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1answer
77 views

$\lim S_n$ finite implies $\sum E(X_n)$ finite

I am learning Second Borel-Cantelli Lemma now and come across a problem. If $X_n \in [0,1]\forall n$,$S_n=\sum^n_{i=1}X_i$,$X_i$ independent and $\lim S_n<\infty$, then $\sum EX_n$ is finite. ...
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0answers
12 views

Find the expectation of a function that contains absolute value of Gaussian variables?

Given $x \sim \mathcal{N}(x|0;1)$ and $y \sim \mathcal{N}(y|1;1)$,$x,y$ are two independent variables. How to find the expectation $\int \int (1+|x-y| )e^{-|x-y|}\mathcal{N}(x|0;1)\mathcal{N}(y|1;1) ...
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3answers
53 views

Addition of two Binomial Distribution

What is the distribution of the variable $X$ given $$X=Y+Z$$, where $Y$~Binomial($n$, $P_Y$) and $Z$~Binomial($n$, $P_Z$)? For the special case, when $P_Y = P_Z = P$, I think that X~Binomial($2n$, ...
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0answers
10 views

The variance of a multivariate normal random variable

Suppose $\vec{X}$ is an N-dimentional random vector that is multivariate normal distributed: $\vec{X} = [X_1, X_2, ..., X_n]^T$ and $X_i \sim N(0,s_i^2)$ and all correlations bewteen $X_i$ and $X_j$ ...
1
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1answer
61 views

Conditional probability of Poisson given Y1 + Y2 = m

I had a probability question I was hoping for some help on! First, the question: Let Y1 and Y2 be independent Poisson random variables with means λ1 and λ2, respectively. a) Find the ...
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0answers
23 views

Linear combination of Chi-squared distrubuted variables with ascending degrees of freedom

If we have i.i.d. random variables$ \quad X_1,\dots , X_n, \ \text{where} \ X_k \sim \mathcal{N} (\mu_k,\sigma_k^2),$ $\quad$ then $$ Y =\sum_{k=1}^n a_k X_n \sim \mathcal{N} (\sum_{k=1}^n a_k ...
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0answers
17 views

Limit with Cauchy density and arctan

Suppose $\{X_i\}$ is an iid sequence of standard Cauchy variables, i.e each has pdf $\dfrac{1}{\pi(1+x^2)}$ for $x\in \Bbb{R}$. Now I have to show that $$\lim_{n\to \infty}\Pr(X_{(n)}\le nx)$$ exists ...
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1answer
17 views

What is the average wait time in this trivial example?

Suppose we are at a noodle place, there are n seats and each patrons eat noodle within a time of S seconds. Suppose that all the n patrons start to eat at the same time, and finish at the same time. ...
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1answer
30 views

Understanding pseudocode for binomial RV generation

I am trying to understand the following pseudocode used to generate a binomial RV with parameters n,p. I'm not sure what is going on, and I would appreciate if someone could describe what is ...
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1answer
28 views

MODE of mixed distribution

Suppose $f_X(x) = \left\{\begin{matrix} 0.5 & x = 0 \\ x & 0 < x \le 1 \end{matrix}\right.$ What is the MODE of this distribution? I think it should be $0$ coz it has the highest mass, ...
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1answer
39 views

Showing that $\left\{ {\mathop {\lim }\limits_{n \to \infty } {X_n} = X} \right\}$ is an event

Let $X$ and ${\left( {{X_n}} \right)_{n \in \mathbb{N}}}$ be random variables on a measurable space $\left( {\Omega ,\mathcal{F}} \right)$. Show that: 1) $\left\{ {\omega \in \Omega :\mathop {\lim ...
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1answer
45 views

grading problem Probabilty

I was practicing my probability and came across this question which i couldn't solve. A particular professor is known for his arbitrary grading policies. Each paper receives a grade from the set {A, ...
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0answers
26 views

Ornstein-Uhlenbeck -> AR(1)

Any suggenstions to where I can read a rigorous proof of how the descrete time version of an Ornstein Uhlenbeck process can be considered as an AR(1) process?
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1answer
40 views

Random variable which is convergent to $0$ but with mean $\infty$

I have problems with understanding the following example: Suppose $\left( \Omega, \mathcal{F}, \mathbb{P}\right)=\left([0,1], \mathcal{B}([0,1]) , \lambda|_{[0,1]}\right)$ and the sequence of random ...
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1answer
23 views

Need little clarification about Neyman-Pearson Lemma

According to my text-book Neyman-Pearson lemma says that the most powerful test of size $\alpha$ for testing point hypotheses $H_0: \theta=\theta_0$ and $H_1: \theta=\theta_1$ is a likelihood ratio ...
3
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3answers
73 views

Correlation of uniform variables

Let $X$ and $Y$ be independent random variables, $X,Y \sim unif(0,1)$. Let $U = \min \{X,Y\}$ and $V = \max\{X,Y\}$. Find the correlation coefficient of $U$ and $V$. I think we can assume that $U = ...
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3answers
20 views

Finding covariance of $X+Y$ and $X-Y$, where $X$ and $Y$ normal RV's

How would I find the covariance of $X+Y$ and $X-Y$, given that $X$ and $Y$ are independent normal random variables, both with mean $0$ and variance $1$? My attempt: ...
2
votes
1answer
25 views

Find the expectancy of $X$

Let $p,q \in (0,1)$. Let $Y$ be the R.V denotes the number of days of the storm in the ocean. $Y\sim \text{Bin}(n,p)$. Let $X$ be the number of ships drowned during the storm and we know that the ...
0
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1answer
17 views

Are linearly dependent random variables necessarily correlated?

'Random variables that are uncorrelated are not necessarily independent, since they can be dependent in non-linear ways' So does this mean that a linearly dependent random variables are necessarily ...