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

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10
votes
3answers
4k 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 ? ...
10
votes
1answer
142 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 ...
9
votes
1answer
443 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 ...
9
votes
2answers
748 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
1answer
243 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 $(X_n)_*(\mathbb{P})$ ...
7
votes
3answers
257 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
227 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
2answers
748 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
0answers
58 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. ...
6
votes
2answers
2k 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 $\mathbb{E}[X] = +\infty$?
6
votes
4answers
298 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
1answer
52 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 ...
6
votes
0answers
67 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 ...
6
votes
0answers
170 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 ...
6
votes
1answer
115 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
225 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 ...
5
votes
3answers
73 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? ...
5
votes
2answers
496 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 ...
5
votes
2answers
75 views

Expectation with square root

I don't know how to calculate the expectation when there is some square root in the expression. My problem is this: we have three real random variables $X,Y,Z$, independent and with standard normal ...
5
votes
4answers
593 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 ...
5
votes
5answers
755 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 ...
5
votes
1answer
112 views

Finding E(x) from E(ln(x)).

Say you have $\operatorname{E}[\ln(x)]=\mu$, is there a way to find $\operatorname{E}[x]$? This seems like a really simple question but I can't figure it out. Any help would be appreciated.
5
votes
1answer
305 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 ...
5
votes
1answer
4k views

Generate Correlated Normal Random Variables

This will be a difficult question to explain, but I'll give it my best. I'm running a simulation with a group of objects (let's just call them agents) and each agent has $n$ parameters that defines ...
5
votes
3answers
83 views

Bernoulli Random Variables and Variance

The question is: Suppose $Z_1, Z_2, \ldots $ are iid $\operatorname{Bernoulli}\left(\frac{1}{2}\right)$ and let $S_n = Z_1 + \ldots +Z_n$. Let $T$ denote the smallest $n$ such that $S_n = 3$. ...
5
votes
2answers
702 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.
5
votes
1answer
153 views

Showing that $X\sim E[X\mid\mathcal{G}]$ implies $X=E[X\mid\mathcal{G}]$ almost surely

Suppose $(\Omega,\mathcal F,P)$ is a probability space and that $\mathcal G$ is a sub-sigma-algebra of $\mathcal F$. If $X$ is an integrable, non-negative random variable with the same distribution as ...
5
votes
2answers
89 views

Variance of the sum of Bernoulli Random variables?

$\newcommand{\var}{\operatorname{var}}$ Let $X_{i}$ be a Bernoulli random variable with paramater $p_{i}$ where $p_i$ itself is a random variable that ranges from $0$ to $1$. The expectation of ...
5
votes
1answer
145 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 \, \, ...
5
votes
1answer
217 views

Prove that expected value of X is greater than Y, if given that $P(X\ge Y)=1$

I have to prove that $E(X)$ (Expected Value of a random variable X), is greater than $E(Y)$, if given that $P(X\ge Y)=1$. my thoughts so far: I know from the $P(X\ge Y)=1$ statement, that the values ...
5
votes
1answer
121 views

Density function of $\max(X_1,\dots,X_n)$.

I'm making this statistics exercise and I'm not sure about my solution. Find the density function of $Y=\max(X_1,\dots,X_n)$ if they are all i.i.d. This was my take on this question: $F_Y(a)=P(X_1 ...
5
votes
1answer
81 views

Closed formula for mean

Suppose we have the i.i.d. random variables $X_{11}, X_{12},\ldots, X_{nn}$, such that each $X_{ij}$ has standard normal distribution $N(0,1)$, with mean $0$ and variance $1$. Given some integer ...
5
votes
1answer
186 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
5
votes
0answers
167 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
0answers
114 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le ...
5
votes
2answers
272 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 ...
4
votes
2answers
121 views

Where does this equality about expectation of random variables come from?

In order to prove this Lemma in my course about Probability : Let $X=(X_1,\dots ,X_p)$ be a gaussian random variable such that $\mathbb{E}[X_j]=0$ for all $j=1,\dots,p$. Then ...
4
votes
2answers
80 views

Transformation on a random variable

Can someone please help me with formatting this question? $Y$ is an exponential random variable with parameter $1$. Let $Z=-Y$, what is the pdf of $Z$? Attempt: $$\Pr(-Y< y)=\Pr(Y>-y) ,$$ ...
4
votes
2answers
264 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
4
votes
2answers
8k 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 $X$.
4
votes
1answer
1k 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. ...
4
votes
1answer
250 views

What is this operation on random variables called?

Let $X$ be a random variable and let $N$ be a discrete random variable which takes values in the non-negative integers. Let $X_1, X_2, ...$ be a sequence of i.i.d. random variables with the same ...
4
votes
2answers
47 views

Conditional Expectation of X given X^2

What can we say about $E[X|X^2]$ in general? And if $X$ has density $f$ respect the Lebesgue measure?
4
votes
1answer
130 views

Understanding the definition of a random variable

I'm working through a math stats book on my own (I've always wanted to learn it), but I'm getting confused about the definition of a random variable. The book says that a random variable is a ...
4
votes
4answers
1k 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 ...
4
votes
2answers
87 views

Eigenvalues of a Random Matrix

I am studying the theory of random matrices lately, but there is a basic issue troubling my life. I hope someone here explain me this, thank you. A random matrix is defined as a matrix whose entries ...
4
votes
2answers
107 views

special sum of binomials distributions

Let $X$ be a random variable. Let $X_p$ be distributed as a Binomial distribution with number of outcomes $X$ and probability $p$, i.e. $Bin(p, X)$. Consider the random variable, $$ Y = X_p + X_{1-p}. ...
4
votes
1answer
59 views

The repetition pattern of a random integer sequence

Here is a problem that bothers me, could some one grand me some help? There is a sequence of N random integers, {$X_1,X_2,...,X_N$}. Each $X_i$ is uniformly chosen from a integer set {1,...,M}. For ...
4
votes
2answers
104 views

Measurability problem of sample distribution function of a contiuum of independent random variable

Let $I = [0,1]$ be the index set of a contiuum of i.i.d random variables. For each $t \in I$, the sample space of $X_t$ is $\Bbb R$ equipped with Borel $\sigma$-algebra and Borel probability measure. ...
4
votes
1answer
102 views

How to find $\mathbb{E}[X\mid\min(X,Y)]$?

Say I have two independent variables $X$ and $Y$ that are exponentially distributed with respective rates $\lambda_X$ and $\lambda_Y$. How do I compute $\mathbb{E}[X\mid \min\{X,Y\}]$?