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9 votes
Accepted

Jensen's inequality for random functions in a Banach space

Here is an example where Jensen’s inequality $f(E[\mathbf{X}])\leq E[f(\mathbf{X})]$ for convex functions $f$ fails when $f$ is defined over an infinite dimensional vector space, even when all ...
Michael's user avatar
  • 24.9k
6 votes

What is the expectation of norm of $[X_1,\ldots, X_n]$ where $X_i$ are indpendent complex Gaussian random variables

EDIT: expanded original answer by analyzing the base case, and an extra explanation on the original answer. Using normalization, each $Y_i=\frac{X_i}{\sigma_i}\sim\mathcal{CN}(0,1)$. Hence, $|Y_i|^...
cjferes's user avatar
  • 2,236
6 votes
Accepted

In algebra of random variables is $X + X \neq 2X$?

Consider the random variable $X$ with the following pmf: $$\mathbb{P}(X=k)=\begin{cases}1/2&\text{for }k=1\\ 1/2&\text{for }k=0.\end{cases}$$ If $X_{1},X_{2},X$ are i.i.d., then let's compute ...
RideTheWavelet's user avatar
4 votes
Accepted

Conditional expectation of random function

Consider the probability space $$[\Omega,\mathcal A, P]$$ and let $\mathcal G\subset \mathcal A$ be a $\sigma$-algebra. Let $f$ be a simple function for any fixed $x$, that is, let $$f(x,\omega)=f_i(...
zoli's user avatar
  • 20.5k
4 votes
Accepted

Function Composition and Expected Value

If we take a given number $x_0$, then our random number is a uniform random variable $X$ on $[0,x_0]$. Its probability density function is: $$ f_{X}(x) = \begin{cases} \frac{1}{x_0} & 0\leq x \leq ...
Manuel Guillen's user avatar
4 votes
Accepted

Projections of uniformly distributed $\mathbb{R}^3$ unit vector have uniform distribution

This was proved by Archimedes and has become known as "Archimedes' hat box theorem." They proved that if a sphere is inscribed in a vertical cylinder, the area of the sphere between two horizontal ...
Dap's user avatar
  • 25.3k
4 votes
Accepted

Can a function that selects between two random variables increase the variance more than twofold?

Let $A_1, A_2$ be the disjoint events $\{f=1\}, \{f=2\}$. Then we can write $$Y = 1_{A_1} X_1 + 1_{A_2} X_2.$$ Note that $$Y^2 = 1_{A_1} X_1^2 + 1_{A_2} X_2^2$$ Now $$\begin{align*} \operatorname{Var}...
Nate Eldredge's user avatar
4 votes
Accepted

What is the Borel sigma-field of positive functions?

As you say, no special topological hypothesis is made concerning $X$, so one must assume that by $B(S,\Bbb R^+)$ the author intends the product $\sigma$-field on $(\Bbb R^+)^X$. As the author intends ...
John Dawkins's user avatar
  • 26.6k
4 votes

If $X_t = Y_t$ in distribution, for any $t \in T$ (compact), is it true that $\mathbb E \sup_{t \in T} X_t = \mathbb E\sup_{t \in T} Y_t$?

Let $T= {1, 2}$ and $X_t, Y_t$ be symmetric Bernoulli random variables. Let $X_1$ and $X_2$ satisfy $P(X_1X_2=1)=1$, and $Y_1$ and $Y_2$ satisfy $P(Y_1Y_2=-1)=1$. Then we have $P(\max_{t \in T}X_t=1)=...
I H's user avatar
  • 1,177
3 votes
Accepted

First approximation of the expected value of the positive part of a random variable

Special case 1: If $X\sim N(0,\sigma^2)$ then $$E(X^+)=\int_{0}^{\infty} x\cdot \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{x^2}{2\sigma^2}}dx = \sqrt{\frac{1}{2\pi}} \sigma.$$ Special case 2: If $X$ ...
rookie's user avatar
  • 1,738
3 votes

what are the characteristics of this random variable?

$\frac{1}{n} \sum_{i=1}^n r_i \sim N(\theta, \sigma^2/n)$ (why?) $\frac{1}{n} \sum_{i=1}^n r_i + \rho \sim N(\theta, a^2 + \sigma^2/n)$ (assuming $\rho$ is independent of the $r_i$) $\sqrt{b} \left(\...
angryavian's user avatar
  • 90.8k
3 votes

Projections of uniformly distributed $\mathbb{R}^3$ unit vector have uniform distribution

Hint: In this answer it is shown that the area of the green region on the sphere is the same as the area of the red region on the cylinder. The area of the red region on the cylinder is $2\pi$ times ...
robjohn's user avatar
  • 347k
3 votes

Projections of uniformly distributed $\mathbb{R}^3$ unit vector have uniform distribution

This property is an immediate consequence of the fact that the "horizontal projection" of $S^2$ onto a cylinder of height $2$ enveloping $S^2$ along the equator is rotationally symmetric with respect ...
Christian Blatter's user avatar
3 votes
Accepted

Picking a special function from the set of random functions

First note that $\ f(0,x)-f(1,x)\in\{-1,0,1\} \ $. If $\ b\in\{0,1\}\ $ and $\ q\ge3\ $, then $$ f(0,x)-f(1,x)\equiv b\text{ mod}(q)\iff f(0,x)-f(1,x)= b\ , $$ so the probability you're looking for ...
lonza leggiera's user avatar
3 votes
Accepted

Continuity of sup-norm for random functions

Assume $W$ continuous $K\to R$ and first assume $K$ convex. Let $M_\epsilon(t) = \sup_{s:|t-s|\le \epsilon} |W(t)-W(s)|$. Since the intersection of $K$ with the closed ball around $t$ is compact, ...
jlewk's user avatar
  • 1,877
2 votes
Accepted

If $U$ has uniform distribution over the interval $(0, 1)$ what is the density function of $X = -Kln(U)$ for some constant $K > 0$?

We start by looking at the cumulative distribution function for $X$, \begin{equation} F_X(x) =P(X\leq x) =P(-K\ln U\leq x) = P(\ln U \geq -\frac{x}{K})=P(U\geq e^{-\frac{x}{K}}) \end{equation} Now we ...
Jürgen Sukumaran's user avatar
2 votes
Accepted

what are the characteristics of this random variable?

Since $r_i \sim \text{IID } \mathcal{N} (\theta, \sigma^2)$ you have: $$\bar{r}_n \equiv \frac{1}{n} \sum_{i=1}^n r_i \sim \mathcal{N} \Big( \theta, \frac{\sigma^2}{n} \Big).$$ Since $\rho \sim \...
Ben's user avatar
  • 4,149
2 votes

Will operations on a purely random function also be random?

The values of random will be uniformly distributed in the interval $(x,y)$. I think that's what you mean by "completely random". But the values of your function ...
Ethan Bolker's user avatar
  • 96.8k
2 votes

Integral with respect to random measure is measurable

No, your argument is not complete. Given $g$ there exist simple functions $g_n,n=1,2,...$ such that $g_n \to g$ uniformly. $\int g_ndP(\omega)$ is measurable for each $n$. To show that $\int gdP(\...
Kavi Rama Murthy's user avatar
2 votes
Accepted

What does this vector function notation mean?

By writing $X = [X_1, X_2]^T$ you are limiting the discussion to $n = 2$, so I'll keep that way. As they said, $H$ is the inverse of $G$. It is a function from $\mathbb{R}^2$ to $\mathbb{R}^2$, which ...
Integral's user avatar
  • 6,574
2 votes
Accepted

Explanation of $\lim\sup$ of a sequence of random variables in measure theory

Two random variables having the same distribution does not mean they take the same value on each $\omega$. Indeed, independence already kill that possibility unless the random variable is constant (a....
user10354138's user avatar
  • 33.3k
2 votes
Accepted

Multiplication of random matrices with independent entries

The entries of $AB$ need not be independent. As an example, let $A,B$ be $2{\,\times\,}2$ matrices with entries chosen independently from a uniform distribution on $\{-1,0,1\}$. For two such random ...
quasi's user avatar
  • 59.1k
2 votes
Accepted

Convergence in law for a family of random generalized functions

Yes. The convergence in law or distribution for random variables, in general, means the weak convergence of their laws or probability distributions. For this last notion you need to have a fixed ...
Abdelmalek Abdesselam's user avatar
2 votes
Accepted

An Application Kolmogorov's Three Series Theorem

Write $$0 = E X_i = E[ X_i 1_{|X_i| \leq 1}] + E[ X_i 1_{|X_i| > 1}] = E[Y_i] + E[ X_i 1_{|X_i| > 1}]. $$ Thus $$|E[Y_i]| \leq E[|X_i| 1_{|X_i| > 1}] \leq E[\psi(X_i)].$$
Marcus M's user avatar
  • 11.2k
2 votes
Accepted

Probability that quadratic polynomial in independent gaussian variables is negative

We can rewrite the expression as $(w\cdot a)(w\cdot b)$. This is positive when the dot products of $w$ with $a$ and $b$ are both positive or both negative. We can normalize $a$ and $b$ to unit vectors....
Anand's user avatar
  • 1,236
2 votes
Accepted

Events of correlated jointly Gaussian distributions

I have found a rate of convergence that is sufficient to me. It is the case that $$F(\rho):=\mathbb{P}(\{|X|>1\}\cap\{|Y|>1\})-\mathbb{P}(\{|X|>1\})^2\leq L|\rho|$$ for some constant $L>0$....
courageousmartingale's user avatar
2 votes

Relation Between Subgradients of a Random Function and Its Expectation

Since, $$f(u,y) \geq f(x,y) + g(x,y)^T(u-x)$$ we have that, $$E_Y(f(u,y)) \geq E_Y(f(x,y) + g(x,y)^T(u-x))$$ $$E_Y(f(u,y)) \geq E_Y(f(x,y)) + E_Y(g(x,y))^T(u-x)$$ So $E_Y(g(x,Y))$ is a sub-gradient. ...
Balaji sb's user avatar
  • 4,393
2 votes
Accepted

Birthday problem: how to show the scaling with $1/N^2$?

You can do this using a sort of continuous version of the combinatorics technique sometimes called "stars and bars". Let's just take the $x_k$ to be $N$ independent random numbers chosen ...
Gareth McCaughan's user avatar
1 vote
Accepted

Linear combination of non-identically distributed, independent exponential random variables

Using the moment generating function: $$M_{\hat{\beta_1}}(t) = E[e^{\hat{\beta_1}t}] = E\left[\prod_{i=1}^ne^{k_i Y_i t}\right] = \prod_{i=1}^n E[e^{k_i Y_i t}] = \prod_{i=1}^n M_{Y_i}(k_i t) = \...
Andres's user avatar
  • 53

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