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

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19
votes
3answers
10k views

Expectation of the min of two independent random variables?

How do you compute the minimum of two independent random variables in the general case ? In the particular case there would be two uniforms variables with difference support, how should one proceed ? ...
16
votes
2answers
857 views

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables ...
14
votes
2answers
1k views

Conditional expectation on more than one sigma-algebra

I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two ...
11
votes
1answer
4k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
11
votes
1answer
246 views

Why is the function $\Omega\rightarrow\mathbb{R}$ called a random variable?

I do not understand the relation of a normal variable "x", which is to me just a placeholder for an element of a set, and a random variable, which is a mapping from the set of all possible events to ...
11
votes
1answer
16k views

Computing the Expectation of the Square of a Random Variable: $ \text{E}[X^{2}] $.

What is the rule for computing $ \text{E}[X^{2}] $, where $ \text{E} $ is the expectation operator and $ X $ is a random variable? Let $ S $ be a sample space, and let $ p(x) $ denote the probability ...
11
votes
1answer
1k views

Uniform distribution on a simplex via i.i.d. random variables

For which $N \in \mathbb{N}$ is there a probability distribution such that $\frac{1}{\sum_i X_i} (X_1, \cdots, X_{N+1})$ is uniformly distributed over the $N$-simplex? (Where $X_1, \cdots, X_{N+1}$ ...
10
votes
0answers
174 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
9
votes
1answer
5k views

Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
9
votes
2answers
15k views

Convex Combinations of Low Probability Bernoulli Variables

Let $X_1,\dots,X_n$ be independent Bernoulli variables with probability $p<\frac{1}{2}$ (even $p\le\frac{1}{3}$ if needed). Let $\alpha_1,\ldots,\alpha_n$ be non-negative real numbers such that ...
9
votes
0answers
332 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
8
votes
2answers
5k views

Infinite expected value of a random variable

How can a positive random variable $X$ which never takes on the value $+\infty$, have expected value $\mathbb{E}[X] = +\infty$?
8
votes
7answers
2k views

“Random” generation of rotation matrices

For a current project, I need to generate several $3\times 3$ rotation matrices for input into an algorithm. I thought I might go about this by randomly generating the number of elements needed to ...
8
votes
2answers
516 views

Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I ...
8
votes
1answer
395 views

Weak convergence of random variables

Suppose we have an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sequence of random variables $X_n:\Omega\to\mathbb{R}$ such that the pushforward measures $(X_n)_*(\mathbb{P})$ ...
8
votes
2answers
571 views

Existence of iid random variables

In probability theory we often used the existence of a sequence $(X_n)_n$ of independent and identically distributed random variables. This was already discussed here. One of the answers says: As ...
8
votes
2answers
2k views

Random sum of random variables

Say you sum i.i.d. variables $X_i$ a total of $Y$ times. If you know the distribution of random variables $Y$ and $X_i$, what is the calculation you have to do to get the distribution of the sum?
7
votes
4answers
6k views

What does it mean to integrate with respect to the distribution function?

If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as: $$E(X) = \int x f(x) dx$$ where the ...
7
votes
4answers
9k views

Relationship between Binomial and Bernoulli?

How should I understand the difference or relationship between Binomial and Bernoulli distribution?
7
votes
4answers
3k views

Question on the 'Hat check' problem

The famous 'Hat Check Problem' goes like this, 'n' men enter the restaurant and put their hats at the reception. Each man gets a random hat back when going back after having dinner. The goal is to ...
7
votes
1answer
6k views

Mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$?

What is the mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$? It is interesting to note that Gaussian R.V here is zero-mean and non-central Chi-square Distribution ...
7
votes
2answers
17k views

Probability distribution of a sum of uniform random variables

Given $$X = \sum_i^n x_i$$ ,where $x_i \in (a_i,b_i)$ are independent uniform random variables, how does one find the probability distribution of $X$.
7
votes
1answer
211 views

Mean value theorem for random variables (inside an expectation value)

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$E[f(X+Y)]=E[f(X)+E[f^\prime(X+\theta Y)]Y]$$ for real valued random variables $X$ and $Y$ ...
7
votes
1answer
100 views

Finding tight upper/lower bounds for $\mathbb{E}[\frac{1}{1+X^{2}}]$ where $X$ is a RV with $\mathbb{E}[X]=0$ and $\mbox{Var}(X)=\nu<\infty $

The question is pretty much in the title. My first thought was using Jensen's inquality to get some sort of lower bound. Since $\frac{1}{1+x^{2}}$ is convex on ...
7
votes
3answers
286 views

Probability that a vertex in the spanning tree of an $N$ x $N$ grid graph is a leaf

Suppose we have an $N$ x $N$ grid graph $G(V,E)$ and we construct a spanning tree of this graph in the following way. Start with a set $S$ which contains only the vertex at the top left corner of the ...
7
votes
1answer
88 views

A.s. equality between limsup of random variables

"Let $(X_n)_{n\ge 1}$ be a sequence of uniformly bounded random variables defined on a probability space $(\Omega, \mathscr{F}, P)$. Moreover define $\mathscr{F_0}=\{\emptyset,\Omega\}$ and ...
7
votes
1answer
522 views

Laplace transform of a random variable

My professor says that the Laplace transform of a nonnegative RV uniquely determines the RV up to distributional equality among all nonnegative RVs. He says one can argue this by appealing to a fact ...
7
votes
0answers
96 views

Existence of Radon Nikodym derivative of Stieltjes measures

Let $X$ be a real valued random variable and let $Q_{X}$ be a right-continuous quantile function for $X$ (alternatively this is a right-continuous generalized inverse to the CDF of $X$ ). ...
7
votes
0answers
230 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...
7
votes
0answers
182 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
7
votes
0answers
401 views

What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
6
votes
2answers
1k views

Summing (0,1) uniform random variables up to 1 [duplicate]

Possible Duplicate: choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries? So I'm reading a book about ...
6
votes
4answers
444 views

What exactly is a random variable?

I don't really understand the definition of a random variable. I also find the wikipedia entry on random variables kind of confusing. Can someone give me a clear explanation of the random variable?
6
votes
3answers
4k views

Difference of two binomial random variables

Could anyone guide me to a document where they derive the distribution of the difference between two binomial random variables. So $X \sim \mathrm{Bin}(n_1, p_1) $ and $Y \sim \mathrm{Bin}(n_2, p_2) ...
6
votes
1answer
13k views

Generate Correlated Normal Random Variables

I know that for the $2$-dimensional case: given a correlation $\rho$ you can generate the first and second values, $ X_1 $ and $X_2$, from the standard normal distribution. Then from there make $X_3$ ...
6
votes
2answers
185 views

Are random selections from i.i.d. random variables independent?

Let us have identically independently distributed random variables $x_1, x_2, \dots, x_{10}$. Now let us pick indices $\alpha, \beta$ uniformly independently from $1,2,\dots,10$. Are variables ...
6
votes
1answer
1k views

Jensen's Inequality (with probability one)

In the following theorem, I have a problem about the second part. That is showing if $f$ is strictly convex then $X=EX$ with probability $1$. While I can see this must be true, I don't know how to ...
6
votes
1answer
60 views

Divergent series of random variables

I've been trying to prove that given a sequence of independent random variables with identical distribution $\{X_n\}_{n \in \mathbb{N}}$ such that $P(X_1 \neq 0)>0$, so also $P(X_i \neq 0) >0 \ ...
6
votes
2answers
3k views

What is the PDF of random variable Z=XY?

Given two independent random variables X and Y, how can I find the PDF of random variable $Z=XY$? *If their joint distribution is required, assume that we also have it.
6
votes
1answer
246 views

Gambling Game: Martingales

This is a multipart question; if there's a strong preference for breaking this into separate questions I'll do that. Imagine a game between a gambler and a croupier. Total capital in the game is ...
6
votes
1answer
78 views

Is this set of random variables a Hilbert space?

Consider a sequence of i.i.d. random variables $\left\{ {{\varepsilon _t}} \right\}_{t = 1}^\infty $ with $E\left( {{\varepsilon _t}} \right) = 0$ and $E\left( {\varepsilon _t^2} \right) = {\sigma ...
6
votes
1answer
66 views

Trying to understand the behaviour of i.i.d.

In a course called introduction to probability theorem we are covering now i.i.d. (independent and identically distributed random variables). I already know when two variables are independent: $X, Y$ ...
6
votes
2answers
261 views

$\{X_n\}$ are iid random variables with symmetric distribution

Let $X_1,X_2,\ldots,X_n$ be iid random variables with symmetric distribution. Show that $$P\left(|X_1+X_2+\cdots+X_n|\ge \max_{1\le i\le n}|X_i|\right)\ge \frac12.$$ I was trying it for $n=2$. ...
6
votes
1answer
204 views

A consequence of the law of large numbers

Let $(X_k)_{k=1}$ be Poisson random variables with expectation $\mu$, let $Y_n = \sum_{k=1}^{n} X_k$. The weak law of large numbers states that, $$ \forall \delta>0, \forall \epsilon>0 \, \, ...
6
votes
2answers
110 views

Demystifying definition of the Random Variable

The intro on Random Variables says that it is a variable (in bold), whose value depends on a chance. IMO, it sounds like a random value generator, whose value depends on a chance, just as random ...
6
votes
0answers
74 views

Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R $ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
6
votes
0answers
114 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
6
votes
1answer
239 views

Sum of average reciprocal of which random variable converges to a Cauchy distribution?

If $(X_n)_{n\in\mathbb{N}}$ are independent identically distributed random variables with density $f$ even, continuous in $0$ and such that $f(0)>0$, then $$\frac{1}{n}\left(\frac{1}{X_1}+\dots + ...
6
votes
0answers
306 views

expectation value for minimum distance between random variables

note: edited to clarify boundary issue Suppose $x_i$, $i=1\dots n$, are randomly drawn from a uniform circular distribution between 0 and 1 (using periodic boundaries). Let $d_i$ be the distance ...
5
votes
3answers
82 views

Why does $\mathbf{Var}(X) = \mathbf{Var}(-X)$ for random variable $X$?

Question from UCLA Math GRE study packet, Problem Set 2, Number 4: http://www.math.ucla.edu/~cmarshak/GREProb.pdf Let $X$ and $Y$ be random variables. Which of the following is always true? ...