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

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0answers
7 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 ...
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2answers
50 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 ...
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1answer
14 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
26 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
21 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 ...
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2answers
54 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
38 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 ...
1
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0answers
80 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
15 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 ...
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2answers
30 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
82 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
68 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 ...
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1answer
37 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 ...
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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 ...
3
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1answer
39 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 ...
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0answers
47 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 ...
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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
19 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
23 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
49 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 ...
1
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1answer
65 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: ...
0
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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 ...
2
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1answer
33 views

Find the probability generating function $G(s)$ of this branching process.

Suppose that $X_n$ is size of the $n$th generation of a branching process started from a single individual, where each individual has a random number of children with probability mass function: ...
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1answer
27 views

Suppose that $N$ is an iid geometric RV and $X_i$ is an iid Bernoulli RV. Find the p.g.f. of $R=X_1+ \dots + X_n$.

Each year a tree of a particular type flowers once and produces a random number $N$ of flowers, where $\mathbb{P}(N=n)=(1-p)p^n$, $n=0,1,2,\dots $ and $0<p<1$. Each flower has probability $1/2$ ...
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1answer
36 views

Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :

$$P(X_1 < X_2 < X_3)= \frac{(1-p_1)(1-p_2)p_2p_3^2}{(1-p_2p_3)(1-p_1p_2p_3)}$$ To be frank I do not know where to start with this question, I would like an idea to get me going, or better yet ...
2
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1answer
33 views

Convergence in probability of product random variables

If $Y_n$s converge to constant $c$ in probability & $(X_n)$ is a sequense of random variables, is it true that $X_nY_n- cX_n$ converge to $0$ in probability? How can I prove this? Thanks in ...
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1answer
11 views

Function of Jointly Distributed and Convolution

Looking into the continuous case of the sum of jointly distributed RVs in an example in my textbook and there are a few steps missing that I can't seem to wrap my head around. If $X$ and $Y$ are ...
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2answers
34 views

A stick length 1 is broken into 2 pieces. Let $Z_1$ be the length of the shorter part. Find $EZ_1$

This is used: If $p(x)$ is continuous, then $P\{x \leq X \leq x+ \Delta x \}= p(x)\Delta x+ o(\Delta x), \Delta x\to 0.$ Let $H_1$ be the occurrence that the point at which the stick is broken is in ...
0
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1answer
46 views

Let $X: U(0,3 \pi)$ - Uniform on $(0,3 \pi)$ Find the distribution of Y and the expectation of Y if Y is:

$$Y=\begin{cases} -\sin X , x \in(0, \pi] \\- \frac{1}{2} , X \in[\pi, \frac{3 \pi}{2}]\\ \cos X, X \in [\frac{5 \pi}{2}, \frac{11 \pi}{4}] \\ \frac{3}{4}, X \in (\frac{11 \pi }{4}, 3 \pi) ...
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3answers
37 views

Random independent variables, a question of expected value

The density function of $X$ and $Y$ (two independent variables) are respectively : $$\phi_X(x)=\begin{cases} \frac{1}{2}(1+x) , x \in (-1,1) \\ 0, \text{otherwise} \end{cases}$$ ...
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1answer
24 views

Find the density function of $X$, from the random vector $(X,Y)$ if the PDF of this vector is:

$$\phi(x,y)= \frac{|x|}{\sqrt{8 \pi}}e^{-|x|- \frac{1}{2}x^2y^2}, x,y \in R $$ Now I'm aware I would have to do $$\phi_X(x)=\int_{- \infty}^{+ \infty}\phi(x,y) dy$$, what is confusing me is this ...
2
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1answer
79 views

Questions on Kolmogorov Zero-One Law Proof in Williams

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book: Here are my questions: Why exactly are $\mathfrak{K}_{\infty}$ and ...
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1answer
33 views

Definition of random variable

In some books, they don't define the random variable based on measure theory. Instead, they define as follow (in the book All of Statistics of Larry Wasserman): My question is does this definition ...
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0answers
31 views

Mixed distribution of product of Bernoullie and Gaussian r.v

Confused with the formulation of density function of the following mixed distributed random variable $Z$. $$Z \equiv X \cdot Y,$$ where $( \cdot)$ is product operation, and $X$ and $Y$ being ...
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0answers
42 views

A Gaussian Divided by a Gaussian Equal to A Gaussian Divided by a Constant

I have a neural-network model in which each neuron is associated with an angle $\theta$. Firing rate as a function of $\theta$ is either a Gaussian or a constant. The claim has been made using this ...
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2answers
31 views

A problem on continuous random variables

I was reading a The First course on Probability by Sheldon Ross, while I stuck at this possibly stupid doubt. The problem is : The density function of X is given by $$ f(x) = \begin{cases} 2x, ...
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1answer
41 views

Can anyone help clarifying the geometry in this probability, random variables question.

So basically the question is to find the CDF of $Z$ where $Z$ is the random variable that signifies the distance from a point in a square(sides 1 length) to a fixed vertex of the square. I do not ...
3
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1answer
42 views

Random variable $X$ is given with the density function $ \phi_X (x)= \frac{1}{2} e^{-|x|}$ Find the distribution of the random variable $Y$ if:

$$Y=\begin{cases}-X-2,\ \ \ \ X \leq -1 \\ \ \ \ X, \ \ \ \ \ -1 \leq X \leq 1 \\ \ \ \ \ 1, \ \ \ \ \ \ \ \ \ X >1 \end{cases}$$ Now I'm only interested in $t >1.$ (That is only ...
1
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1answer
33 views

Continuous random variables and probability density function

OK, I know that a random variable $X$ from some probability space to $\mathbb R$, with some additional properties. It is discrete if it's image in $\mathbb R$ is dicrete. It is otherwise called ...
0
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0answers
12 views

Random data generation from a bivariate distribution

Let $X$ and $Y$ be non negative random variables with joint distribution \begin{equation} F(x,y)=1-e^{-x}-e^{-y}+e^{-x-y-\delta xy}; ~~~x\geq 0,~~y\geq 0. \end{equation} How to generate a bivariate ...
1
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2answers
33 views

Proving the $Pr(d>0|a+d=\pi)$ is increasing in $\pi$ when a and d are two independent normal distributions.

I was wondering if it is possible to prove the following (or show false otherwise). Given two independently distributed random variables $a\sim \mathcal{N}(\alpha,\sigma_\alpha^2)$ $d\sim ...
2
votes
1answer
22 views

Conditional probability with max(X, Y)

Let $Y_n=$ the outcome of the $n$-th die roll, let $X_{n+1} = \max \{X_n, Y_{n+1}\}$ with $X_1=Y_1$. What is $P(X_{n+1}=j \ | X_1=i_1, ..., X_n=i)$? I know that it is $P(\max \{X_n, Y_{n+1} \}=j \ | ...