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

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17 views

Notation: how can I say variable drawn from a distribution D lies in space X

Suppose I have a random distribution $D$ for which, if $x\sim D$, then $x\in X$. Is there a standard notation involving only $D$ and $X$? For example, let $N$ be the multinormal distribution with ...
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1answer
30 views

Direction on how to rightly compute a certain expected value.

Dan has 13 different cards, one of which is numbered with a unique number between 1-13. Every day, he pulls a specific card (which he never returns) with a uniform distribution between the cards left. ...
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0answers
20 views

Convergence in probability of function composition.

I need to show that $G_n \stackrel{P}{\to}_n F_0$, i.e. for any $\epsilon>0$ $$ P(|| G_n - F_0||>\epsilon) \to_n 0 $$ We know the following: $G_n$ and $F_0$ are a bilinear functions from ...
2
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1answer
21 views

Derivation of Likelihood Function for Random Effects Parameters

I initially posted this question in CV, but getting no responses or interest, I am deleting it there, and trying my luck in math.stackexchange, hoping that the math details of the following derivation ...
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1answer
31 views

Continuous random variable pdf question.

The continuous random variable X has pdf where $$f(x) = \begin{cases} \frac{25}{12(x+1)^2},\quad & 0\le x\le 4 \\ 0 & \text{otherwise} \end{cases} $$ $E(X+1) = 1\frac23$ and $E(X) = 2/3$ ...
1
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0answers
34 views

expectation value of independent random variables

In the statistics lecture that I'm attending, the professor once used the following: $X, Y$ random variables and i.i.d., then $$\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$$ I was trying to see an ...
1
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1answer
26 views

Exponential(1) distributed random variable convergence

I am stuck with convergency in probability... I have the following exercise: Let $(X_k)_{k\ge1}$ be a sequence of independent exponential-(1) distributed random variables. Show that $n^\alpha ...
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2answers
29 views

Find parameters so that random variables (connected to Brownian movement) are independent.

$W_t\sim\mathcal{N}(0,t)$ is Brownian movement, find values of parameters $a, b$ for which $aW_1-W_2$ and $W_3+bW_5$ are independent. I don't even know where to start, so any hint is highly ...
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1answer
16 views

Find the distribution of some random variable connected to Wiener Process. Please, check my solution.

I need to find a distribution of $ 5W_1-W_3+W_7 $, where $W_t$ stands for Wiener Process $W_t\sim\mathcal{N}(0,t)$. Is this solution right? $E(5W_1-W_3+W_7)=5E(W_1)-E(W_3)+E(W_7)=0$ and since ...
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2answers
36 views

Random variables with common mean, variance and pairwise correlation

Hi I'm currently working through past exam questions and am stuck with the following question: Random variables $X_1$, $X_2$ and $X_3$ are identically distributed, with common mean $\mu$, common ...
2
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3answers
77 views

Does $\operatorname{Cov}(X,Y) = 0$ mean $\operatorname{Cov}(X,\log Y) = 0$?

Suppose $X,Y$ are positive random variables with $\operatorname{Cov}(X,Y)=0$. Define $Z= \log Y$. Does it necessarily follow that $\operatorname{Cov}(X,Z) = 0$? I know it's true for linear ...
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2answers
39 views

Find pmf for $i=0,1,2,3,4$

I have a problem that I'm having trouble with. Here is the problem: "Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players compare their numbers, the ...
0
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2answers
62 views

Show that an expected value of a sum of random variables is finite [on hold]

Let $A_1, A_2,...$ are i. i. d. non-negative random variables, $B_1,B_2,...$ are i. i. d. non-negative variables and $A_1,B_1,A_2,B_2,...$ are mutually independent. We also know that ...
2
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4answers
80 views

What is the statistical equilibrium for this simulation of happy bubbles?

Happy Bubbles I hope this is not too specific or practical, but I just made a simulation of sorts and seem to have hit quite close to an equilibrium (by accident). Now I am wondering if and how you ...
1
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4answers
77 views

Probability formula, a value chosen at random is greater than another chosen value.

Say I have two number ranges, whole numbers only. Range 1: [-3,16] Range 2: [3,22] I choose randomly one number from Range 1 and one number from Range 2. Lets call them x and y. How do I find the ...
1
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0answers
26 views

computing the area of a region using Monte Carlo integration

Suppose that I am interested in estimating the area of $\Gamma \in \mathbb{R}^2$. I do not know the exact shape of $\Gamma$ but I have a sufficiently large number of sample points $(X,Y) \in \Gamma$ ...
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2answers
61 views

A random variable $X$ has support in $[a,b]$, and $\mathbb{E}X=b$. Prove that $P(X\geq\mathbb{E}X)=1$

Let $X$ be a discrete random variable such that $R(X) \subseteq I=[a,b]$, $-\infty<a<b<\infty$. Further let $\mathbb{P}$ be a probability measure. Is it possible to write for $x\in I$: ...
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1answer
38 views

Calculating probability distribution under given constraints

I recently asked a question about the construction of a random variable under given constraints (see: Construct a random variable under given constraints). The only answer to my question suggested a ...
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0answers
20 views

Can I have the following random matrix?

Let $A \in \mathbb{R}^{n \times n}$ a random matrix. Assume $E[A] = I$ and that all $A_{ij}$ are independent. Now let $U \Sigma V^{\top} = A$ be the SVD of A. Let $A'$ be the result of thin-SVD, ...
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0answers
13 views

Reference on the necessity proof of Kolmogorov's three series theorem wihout using central limit theorem.

I want a reference on the necessity proof of Kolmogorov's three series theorem wihout using central limit theorem. I understand some intuition behind taking independent copies of the random ...
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0answers
6 views

Propagating uncertainties in Gaussian fit

I'm doing an analysis where I have a set of random variables with some known uncertainties (the uncertainties are different for each random variable). The random variable is roughly Gaussian ...
1
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2answers
49 views

Construct a random variable under given constraints

In preparation for a probability examination, I am working on the following problem. Problem A box contains three white balls and ten black balls. Balls are drawn without replacement until all the ...
1
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1answer
11 views

Proving a.s. convergence by probabilistic convergence

Consider a sequence of random variable $\{X_n\}$. Let $$A_n = \sup\{|X_k - X_l|: k,l \geq n\}$$ $$B_n = \sup\{|X_k - X_n|: k \geq n\}$$ Now to prove a.s. convergence of $\{X_n\}$, I have seen in a ...
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1answer
22 views

Dominated convergence martingale and uniform integrability

For a fixed $t\in [0,1]$ I have a sequence $(X^t_n)_{n\geq 1}$ of normal distributed random variables which is a martingale and bounded in $L^2$. So by the martingale convergence theorem there exists ...
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1answer
20 views

Question about uncorrelatedness of random variables and distributions

I was wondering, if two random variables are dependent, does that mean that they must be correlated? does one imply on the other or vice versa? Also, if I know that a joint distribution of two ...
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0answers
20 views

Covariance matrix is positive definite, does the variable have density?

I know that if a random $n$-dimensional variable $Y= [Y_, ..., Y_n]^T$ has density $f$, then its covariance matrix $\Sigma $ is positive definite: $\forall x \in \mathbb{R}^n \setminus \{0\}: x^T ...
3
votes
2answers
49 views

Expected value of area of triangle

Here is the problem: Let $A$ be the point with coordinates $(1, 0)$ in $\mathbb R ^2$. Another point $B$ is chosen randomly over the unit circle. What is then the expected value of the area of the ...
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1answer
31 views

Deriving mass/density functions of variables $log(X)$, $X+Y$, $sgn(X-1/2)$.

Could you help me with the following question? Suppose that a point with co-ordinates $(X,Y)$ is chosen uniformly from the square $\{(x,y)\in \mathbb{R}^2: 0 \leq x,y \leq 1\}$. For each of the ...
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1answer
37 views

I.i.d. Discrete Random variables with mean 0 and variance 1 [closed]

Given a sequence of i.i.d. random variables $(X_i)_{i\geq 1}$ such that $\mathbb{E}(X)=0,\ \mathbb{E}(X^2)=1$ consider the sum $S_n=\sum_{k=1}^n{X_k}$. Is it true that independently of the ...
2
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0answers
78 views

Probability inequality given $E(X^2) = 1$ [closed]

This is Exercise 20 on page 198 of Resnick's "A Probability Path". I'm stumped; hints or full solutions appreciated. Suppose $E(X^2) = 1$ and $E(|X|) \geq a > 0$. Prove for $0 \leq \lambda \leq 1$ ...
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1answer
13 views

Variational series , probability.

A Variation series is a series of random variables $Y_1,Y_2...Y_k$ where $k$ represents the $k-th$ largest random variable between $X_1, X_2...$, $X_i$ (independent of eachother , equally ...
1
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2answers
29 views

Sum and difference of three normally distributed variables

We are given three independent random variables $X, Y, Z$ with normal distribution $\mathcal{N}(1,2)$. Are $U=Z-Y+X$ and $V=X+Y$ independent? I thought I would compute the joint density $f_{UV}$ and ...
2
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2answers
80 views

The geometry in this probability question is unclear to me.

On the circle: $x^2+(y-1)^2=1$ at random a point is chosen. Let $C(Z,0)$ be the point in which $0x$ axis and ray $AB, A(0,2)$ interect. Find the distribution function of $Z$. Answer: Let x ...
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1answer
32 views

Density Function $Y=X(2-X)$

Suppose $X$ has density function $\frac{x}{2}$ for $0<x<2$ and $0$ otherwise. Now I am wondering what the density function of $Y=X(2-X)$ will be. I tried to compute $P(Y \geq ...
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0answers
65 views

Confusion about a random process

Let $X(t)$ be a random process such that: $$ X(t) = \begin{cases} t & \text{with probability } \frac{1}{2} \\ 2-at & \text{with probability } \frac{1}{2} \\ \end{cases}, $$ where $a$ is a ...
2
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1answer
33 views

On $[0,1] $ $, 100$ points are chosen at random. $X_1$- number of chosen point between $\frac{1}{5}$

On $[0,1] $ $, 100$ points are chosen at random.(This probably means, uniformly, I can only assume, no other context is given, so let's pressume what seems most natural.) $X_1$- number of chosen ...
1
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3answers
53 views

On the interval $(0,1)$ $9$ points are chosen at random.Let $X$ be the distance from $0$ to the medium between the chosen points.

On the interval $(0,1)$ 9 points are chosen at random.- This most likely means uniformly, I doesn't say more than was is written, just the presumption of choosing these points is what comes to mind ...
4
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1answer
34 views

20 balls are put into 10 boxes, let $X$ be the random variable that accounts for the number of empty boxes. Find $EX$ and $DX$-variance.

What I don't know how to do: Put this into a mathematical model effieciently, I honestly do not know where to start here, I've done problems with balls going into boxes and to find lets say the ...
0
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0answers
46 views

Heteroskesdacity

Consider the following model for real estate values applied to a cross-section of homes: $Price = \beta_0 + \beta_1\cdot SQFT_i + \beta_2 \cdot YARD_i + \beta_3 \cdot POOL_i + \epsilon_i$ where ...
0
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2answers
42 views

Sum of independent random variables is also independent

Given that $X, Y$ and $Z$ are discrete independent random variables, how can one show that $X+Y$ and $Z$ are independent as well? So far, I tried using the definition of independent variables and ...
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2answers
39 views

expected value of $N=\min \{n\in \mathbb N:X_1 +\cdots+X_n\ge1\}$

suppose that for every $i\in \mathbb N$ \begin{equation*}X_i\sim \textrm{Pois }(1)\end{equation*}and \begin{equation*}N=\min \{n\in \mathbb N:X_1 +\cdots+X_n\ge1\}\end{equation*}what is the expected ...
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1answer
18 views

Find distribution mean from the mean and sd of the log

I have a distribution with a long tail and use a model to predict the mean and standard deviation of its log. Given the mean and standard deviation of the log, how do I find the mean of the actual ...
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0answers
20 views

General and sufficient condition of independence

I'm having troubles with this proof: Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean and unit standard deviation. For $(a_0, a_1, ..., a_r)$ a sequence of $r$ real numbers ...
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1answer
28 views

Existence proof for two random variables

I have two probability measures $\nu_1,\nu_2$ on a measurable set $(E,\Sigma)$ and a probability measure $\mu$ on $(E \times E, \sigma(\Sigma \times \Sigma))$ with $$ \nu_1(A) = \mu(A \times E) ...
4
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0answers
41 views

Discrete Time Two sided Gaussian Random Walk : Hitting Time Distribution

I am looking at the hitting time of a two sided Gaussian random walk i.e. $S_{n}=\sum_{i=1}^{n}X_{i}$ where $X_{i}$ are i.i.d normally distributed random variables. The hitting time is ...
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1answer
8 views

Normal RV with mean $1$ and variance $4$ out of standard normal

I have successfully used the Box-Muller algorithm to generate two standard normal random variables. However, my goal is to generate two normal random variables with mean $1$ and variance $4$. Is ...
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1answer
60 views

Strange sum of random variables

So guys, I'm having this hard proof to solve in probability. I don't really know how to tackle it! Hope that someone can help. Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean ...
3
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1answer
35 views

an exercise about changing the measure and convergence in $L^1$

this is exercise 17.12 from probability essentials written by jacod & protter. Suppose $lim_{n→∞} X_n = X$ a.s. Let $Y = sup_n |X_n − X|$. Show $Y < ∞$ a.s. , and define a new probability ...
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1answer
40 views

Let $X: U(0,1)$ and when $X=x$ then $Y:U(\frac{x}{2}, \frac{2x}{3})$ uniform distribution. Find the density function of $Y$ and $EY$

Let $X: U(0,1)$ and when $X=x$ then $Y:U(\frac{x}{2}, \frac{2x}{3})$ uniform distribution. Find the density function of $Y$ and $EY$ I don't know if it would be presumptuousness to say that $Y: ...
1
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1answer
22 views

How do I find $ Pr\{X_1 < k \} $ and $ Pr\{X_1 > k \} $if $X_1 : G(p_1)$- geometric distribution [duplicate]

I would think the song like $1-Pr\{X_1 < k \} $ but what is confusing to me is the fact that this is a discrete random variable, and these inequalities ussually apply to absolute continuous ...