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

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21
votes
3answers
12k 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 ? ...
17
votes
2answers
1k 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 $\{x_1,...
15
votes
1answer
21k 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 ...
15
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 sigma-...
14
votes
2answers
354 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 ...
13
votes
5answers
2k views

Are we guaranteed that the harmonic series minus infinite random terms always converge?

Consider the known harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ and modify it as follows $$\sum_{n=1}^\infty a_n\frac{1}{n}$$ where $$a_n \sim \operatorname{Bern} \left({\frac{1}{2}}\right)$$ i.e. ...
13
votes
2answers
430 views

Is $\pi^k$ any closer to its nearest integer than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to its nearest ...
12
votes
1answer
5k 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. ...
12
votes
1answer
283 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
7k views

Affine 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 $\mathbf{x}$....
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 $X_1, \cdots, X_{N+1}$ ...
11
votes
0answers
194 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) = \sup\{\...
10
votes
1answer
18k 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 $X_3$ ...
10
votes
2answers
647 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 ...
10
votes
0answers
399 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 \...
9
votes
4answers
2k views

Two rifleman shooting at a target where the game ends when two hits are observed.

The problem is as following: Person A and B are shooting at a target. Independently of who is shooting, the probability that the shot results in a hit is $p$, and each shot is independent ...
9
votes
4answers
15k views

Relationship between Binomial and Bernoulli?

How should I understand the difference or relationship between Binomial and Bernoulli distribution?
9
votes
7answers
3k 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 ...
9
votes
2answers
6k 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 $\mathbb{E}[X] = +\infty$?
9
votes
1answer
11k views

How to compute the sum of random variables of geometric distribution

Let $X_{i}$, $i=1,2,\dots, n$, be independent random variables of geometric distribution, that is, $P(X_{i}=m)=p(1-p)^{m-1}$. How to compute the PDF of their sum $\sum_{i=1}^{n}X_{i}$? I know ...
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 $\...
8
votes
4answers
4k 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 ...
8
votes
1answer
8k 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 ...
8
votes
2answers
20k 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 $X$.
8
votes
2answers
8k views

Proof of analogue of the Cauchy-Schwarz inequality for random variables

The Cauchy-Schwarz inequality tells us that for two vectors $u$ and $v$ in an inner product space, $$\lvert (u,v)\rvert \leq \lVert u\rVert \lVert v \rVert$$ with the equality holding iff one vector ...
8
votes
3answers
379 views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
8
votes
2answers
777 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
424 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
1answer
247 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 $Y$ ...
8
votes
1answer
97 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 $\mathscr{...
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?
8
votes
0answers
120 views

Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
8
votes
0answers
218 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
4answers
7k 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
2answers
19k views

how to derive the mean and variance of a Gaussian Random variable?

How do we go about deriving the values of mean and variance of a Gaussian Ransom Variable $X$ given its probability density function ?
7
votes
2answers
17k views

independent, identically distributed (IID) random variables

I am having trouble understanding IID random variables. I've tried reading http://scipp.ucsc.edu/~haber/ph116C/iid.pdf, http://www.math.ntu.edu.tw/~hchen/teaching/StatInference/notes/lecture32.pdf, ...
7
votes
2answers
2k 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 ...
7
votes
4answers
4k views

Convergence in probability of the product of two random variables

Suppose $\{X_n\}$ and $\{Y_n\}$ converge in probability to $X$ and $Y$, respectively. Will $X_n Y_n$ converge in probability to $X Y$? I know the answer is yes. If we treat $(X_n,Y_n)$ as a random ...
7
votes
4answers
491 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?
7
votes
1answer
112 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 $\mathbb{R}\backslash\left[-\frac{1}{...
7
votes
2answers
1k views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
7
votes
2answers
261 views

What's the difference between a random variable and a measurable function?

I've tried to wrap my head around the measure theoretical definition of a random variable for a couple of days now. In his book Probability and Stochastics, Erhan Çinlar defines a measurable function ...
7
votes
3answers
293 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
599 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
1answer
98 views

Gambling system theorem given by Doob

Let$\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variable. Let $\{\alpha_k\}_{k=1}^{\infty} $be a sequence of strictly increasing finite stopping times. Then $\{X_{\alpha_k+1}\}_{n=1}^{\...
7
votes
0answers
90 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 $X$ ...
7
votes
0answers
127 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 ...
7
votes
0answers
459 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
3answers
5k 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
2k 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 ...