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

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0answers
19 views

Central Limit Theorem for independent but non identically distributed random variables

My question is the following: Given the sum of R.V.s, $Z_N = X_1 + X_2 + ... +X_N $, where $X_i$ are independent, Rice distributed ($X_i\sim Rice(\mu_i,\sigma) $), is there any way to approximate ...
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2answers
45 views

Probabilities and random variable problem

Suppose we have a random variable X, and we are given the numerical values of its expectation as well as its s.d. (standard deviation). How can I go about finding the maximum value the probability of ...
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1answer
29 views

What is the meaning of E and d in this formula?

I am trying to learn the information bottleneck method. On slide 15, they give this equation. I think I understand that X is a random variable (but do not understand the meaning of the exponent, n). I ...
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3answers
34 views

Generate random variable from series of its expected values E[X], E[X^2], E[X^3], …?

Given a series of all the expected values of a random variable, can we find the random variable itself ?
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1answer
44 views

Find the cdf associated with each pdf (NOT transformation)

Find the cdc associated with each pdf: a) f(x) = 3(1-x)^2 , 0 < x < 1 , zero elsewhere b) f(x) = 1/x^2 , -infinity < x < infinity The answers are a) 1-(1-x)^3 , 0 <= x < 1 b) 1 ...
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0answers
35 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
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1answer
38 views

Variational problem concerning variances

Let $\phi$ be the family consisting of all random variables $X$ such that $P(X\in [0,1])=1$, $EX=\frac{1}{3}$, $P(X<\frac{1}{4})<\frac{1}{2}$, $P(X>\frac{1}{4})\geq\frac{1}{2}$. Calculate ...
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1answer
21 views

Finding the CDF of $g(X)$ where $X$ is a continuous random variable

I imagine this is a rather simple problem, but I'm having a bit of a hard time actually finding the answer. $X \sim \mathrm{Exp}(0.2)$ and $W=g(X)$ given by $g(X) = \begin{cases} X^{\frac{1}{3}} ...
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1answer
12 views

Two random variables X,Y are, X,Y independant b. are X+Y X-Y independant

if X and Y are independent, check whether the (0, 0) value is the same as P(X=0) P(Y= 0), and the same with the other 4 entries. Make a table with the distributions of X + Y and X - Y. For any ...
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1answer
54 views

A fair die is tossed until the sequence “44” is seen. Let N be the number of tosses this requires. Find E $N$

A fair die is tossed until the sequence “44” is seen. Let $N$ be the number of tosses this requires. Find $E[N]$. I have my own solution which I need someone to verify. And this problem has to be ...
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2answers
43 views

Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
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2answers
131 views

Can the expected value of a PMF be zero, as in E[X] = 0?

The whole question is: Let X be a discrete random variable and let Y = 0.5 X + 3. (i) Assume that the PMF of X is given by where k is some suitable constant. Determine the value of k. (ii) Find E ...
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0answers
7 views

Methods for Uncorrelating data - Comparison

I see that both PCA and Cholesky Decomposition could be used for uncorrelating correlated data. When should one be used? What are the assumptions made by each model. When do the methods fail? Are ...
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0answers
49 views

If $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ can we explicitly define the r.v. $(Y|X\in A)$?

When introducing conditional expectation, one can define $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ which is itself a law. I was wondering if there is a way to define a random variable ...
2
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1answer
31 views

Joint density function with absolute value

Let X and Y have joint density fXY (x, y) = kxy^2 where j0 ≤ x, y ≤ 1, 0 otherwise. Compute Pr(|X − Y | < 0.5). So I found that k=6, but can't figure out the probability part after working on ...
2
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1answer
32 views

Is this a Markov chain? [duplicate]

Let $\{\xi_n \}_{n \geq 1}$ be i.i.d random variables taking values on $\mathbb{Z}$. Let $\xi_0 = 0$. $S_n = \sum\limits_{i=1}^{n} \xi_i,$ where $S_0=0$ $Y_n = \sum\limits_{i=0}^{n} S_i$. My ...
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2answers
31 views

PMF of X: Number of trials to draw a chip

Let a bowl contain 10 chips of the same size and shape. One and only one of these chips is red. Continue to draw chips from the bowl, one at a time and at random and without replacement, until the red ...
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2answers
163 views

PMF of number of heads of 4 coin tosses

Let X equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of X and compute the probability that X is equal to an odd number. I initially ...
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0answers
26 views

Let $X$ be uniformly distributed on $[0,1]$. Find the cumulative distribution function of $X-X^2$.

Let $X$ be a continuous random variable uniformly distributed on $[0,1]$. Find the cumulative distribution function of $X-X^2$. $P(X-X^2 \leq a)= P ( -X^2 + X - a \leq 0) = P ( -X^2 + X - a \leq 0| ...
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2answers
26 views

If $P(X \geq k) = p^k$, for $k=0, 1, 2,…$ then $P(X=k)=p^k(1-p)$

If $P(X \geq k) = p^k$, for $k=0, 1, 2,...$ then $P(X=k)=p^k(1-p)$ The converse is immediate but I don't know how to approach the direct implication.
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1answer
19 views

Transformation of Random Variable results in strange CDF

I'm trying to transform a RV according to $Y=X^{-a}$ with $a>0$ and X being uniformly distributed in $[0,A]$: $ F_X(x) = \begin{cases}0 & x<0 \\ x/A & 0\leq x \leq A \\ 1 ...
2
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1answer
63 views

If $X$ is Poisson, find the expectation of $\frac{1}{a+X}$

If $X$ is a Poisson random variable with $\Pr(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$ and $a>0$ then find the expectation of $\frac{1}{a+X}$ If I make use of ...
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1answer
26 views

How does $\mathcal{L}^1$-convergence of a series of $\mathcal{L}^1$ random variables imply that $\sup_{n \in \mathbb{N}} \mathbb{E}[|X_n|] < \infty$?

Let $(X_n)_{n \in \mathbb{N}}$ be a series of random variables with $\forall i: X_i \in \mathcal{L}^1(\Omega, \mathfrak{F}, P)$ and $X_n \rightarrow^{\mathcal{L}^1}X$. How do I show then, that ...
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1answer
44 views

Random variable modeling arthroscopic meniscal repair

The below problem is from my introductory stats textbook, the chapter on random variables and probability distributions. I don't even know what's being asked, much less how to answer it. Any clues? ...
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1answer
35 views

finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...
2
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1answer
29 views

Finding the mean and median of a probability density function

I suspect this is super-easy, but I haven't done any math in about ten years and I'm working with concepts that have been woefully explained... I need to find the mean and median of a continuous ...
0
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1answer
42 views

How to show that the following equation holds?

In Wikipedia appears the pdf's equation for $XY$ and $X/Y$, where $X$ and $Y$ are given independent random variables. The equations are For product $Z=XY$ ...
3
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2answers
57 views

$X,Y$ independent then $X+Y$, $X-Y$ independent as well?

My question is simple: If $X$, $Y$ are independent random variables then $X+Y$, $X-Y$ independent as well?
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3answers
49 views

Mean and Variance of Y using Expectation Operator

Let $$ Y =\sum_{k=1}^N a_kX_k $$ be the weighted sum of N independent random variables, $ X_k, k = 1, ... , N $ , each having mean $ \mu _{X_i} $ and variance $ \sigma ^2_{X_i} $. The weights $a_k$ ...
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2answers
37 views

Uniform distribution in (0,1). P(X1+X2<=X3) and Gaussian RV with variance 1/4 and 1/9 , P(3V>=2U)

I'm appearing for a competitive examination and I find a lot of questions from probability involving $2$ or more random variables are very common. Please help me with the method on how to deal with ...
3
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1answer
15 views

$L^p$ integrability of products of Gaussian variables

Gaussian variables have moments of all orders, so by Hölder's inequality the product of two Gaussian variables $\xi$ and $\eta$ has finite $L^1$-norm: $$ \|\xi \cdot \eta\|_1 \leq \|\xi\|_2 \cdot ...
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0answers
68 views

Probability that a five is seen before any of the even numbers are seen

A fair die is repeatedly tossed. What is the probability that a five is seen before any of the even numbers are seen? I have my own solution below and just want someone to verify it. According ...
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2answers
50 views

How to work with the mode of a probability mass function

How do you work with a probability mass function in determining stuff related to the mode. Here's the question I have $P(X=x) = {\theta^n}{{n}\choose{x}}({\frac{1-\theta}{\theta}})^x, x = ...
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1answer
63 views

Birthday Problem (Poisson Distribution)

I've been reading up on Poisson Distributions and have come across the following problem. My doubts are in Bold: What's the probability that in a room of n people, nobody shares the same birthday? ...
3
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1answer
28 views

statistics basic question on covariance

anyone would help me in a basic example? a fair coin is tossed, n times. X is the number of Head and Y is the number of Tails. what is the COV(X,Y).
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3answers
35 views

$X$ and $Y$ are independent and follow $U(0,1)$. Show $P(f(X) > Y) = \int_0^1 f(x) dx$

Let $X$ and $Y$ be two independent uniformly distributed r.v. on $[0,1]$, and $f$ is a continuous function from $[0,1]$ to $[0,1]$. Show that $P(f(X) > Y) = \int_0^1 f(x) dx$. I tried to prove ...
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0answers
24 views

Understanding Binomial random variables

I'm looking over Binomial random variables and I understand that $ \sum\limits_{k=0}^n k\binom{n}{k} p^k (1-p)^{n-k} = np $ from $\mathrm{Bin}(n,p)$ However, I don't understand how, if $S_n = ...
2
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1answer
31 views

Understanding Negative Binomial Random Variables

I'm trying to understand Negative Binomial Random Variables and have across the following: $ Z\sim \mathrm{NegBin}(n,p)$ if $Z = X_1 +\cdots+ X_r $ where $X_i's$ are independent identically ...
1
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1answer
13 views

Geometric Random Variable

I'm reading about Geometric Random variables from a book, which is as follows: $X_1, X_2,\ldots$ are independent identically distributed variables which are $\mathrm{Ber}(p)$ $$ Y = \min \{n\geq ...
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1answer
25 views

A LLN type theorem on the supremum of functions of a RV

Let $X_1,\dots,X_n$ be iid real valued random variables. Let $\mathcal{F}$ be a set of functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\mathbb{E}f(X_i) < \infty$ for all $f \in ...
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1answer
28 views

Understanding Geometric Random Variables

I'm looking up Geometric random variables, where $X_1, X_2....$ are independent identically distributed variables which are $Ber(p)$. The book says, $$ Y = \min \{n\geq 1| X_n = 1\} \sim Geo(p)$$ ...
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0answers
59 views

Probability problem related to discrete random variables, binomial distribution.

I've just solved an exercise related to discrete random variables and maybe to the binomial distribution as well. I would like to know if my solution is correct, so here goes the problem statement ...
2
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1answer
61 views

If $Y\ge 0$ almost surely and $X+Y \sim X$ then $Y=0$ almost surely

Let $X, Y$ be random variables on the same probability space such that $Y \ge 0$ almost surely and $X+Y$ and $X$ have the same distribution. Please resolve whether these conditions imply that ...
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1answer
40 views

Conditional expectation of symmetric Sigma algebra

Another exercise with conditional expectation that I have problems with. Let $\Omega=[-1,1]$, $\mathcal{F}=\mathcal{B}(\Omega)$, $\mathbb{P}=\frac{1}{2}\lambda$. Let X be a ...
5
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2answers
43 views

Finding all Borel measures $\mu_X$ such that $Y\sim \mathcal{N}(0,1) \Rightarrow XY \sim \mathcal{N}(0,1)$.

Find all Borel measures $\mu$ on $\mathbb{R}$ such that for every independent random variables such that $X \sim \mu$ and $Y\sim \mathcal{N}(0,1)$ we have $XY \sim \mathcal{N}(0,1)$. To be honest ...
2
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2answers
87 views

Expected distance between two vectors that belong to two different Gaussian distributions

Let $X$, $Y$ be two random variables that follow the Gaussian distributions with mean vectors $\mu_x$, $\mu_y$, and covariance matrices $\Sigma_x$, $\Sigma_y$, respectively. The probability density ...
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2answers
34 views

Prove Kolmogorov's SLLN by martingale.

Suppose $\xi_i$ are i.i.d. and $\mathbb E(|\xi_1|)\lt\infty$ Let $X_n=\sum_{i=1}^n\xi_i$ Then we have $\frac{X_n}{n}\to \mathbb E(|\xi_1|) $a.s. In the proof of this theorem: ...
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0answers
15 views

Simple random walk conditioning on non-return

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0=0$. Let $P_{k,j}$ be the probability that the walker hits the point $k$ without returning to the origin in ...
1
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0answers
35 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
2
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0answers
40 views

Almost sure limit of $\log(X_1 + X_2 + … + X_n) - \log(n)$

Let $X_n$ be an i.i.d. sequence of positive random variables with expectation 2 and variance 1. What is the almost sure limit of $$\log(X_1 + X_2 + ... + X_n) - \log(n)$$ as $n \to \infty$ Would it ...