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

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10
votes
1answer
128 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 ...
10
votes
3answers
3k 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 ? ...
9
votes
2answers
418 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
223 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
419 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 ...
7
votes
2answers
239 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
2answers
683 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?
6
votes
4answers
287 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
199 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 ...
6
votes
0answers
200 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
72 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
1answer
108 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
2answers
1k 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 $\mathbb{E}[X] = +\infty$?
5
votes
2answers
426 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
4answers
425 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
1answer
227 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
3answers
80 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
1answer
148 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
1answer
131 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
176 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
2answers
63 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
0answers
56 views

Covergenge of the sum of reciprocal random variable.

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 + ...
5
votes
0answers
137 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
233 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
119 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
5answers
520 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 ...
4
votes
2answers
189 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
1answer
2k 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 ...
4
votes
2answers
6k 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
238 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
546 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.
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
70 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
104 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
57 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
99 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
825 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
2answers
109 views

The probability P[X < g(Y)]

Suppose that $X$ and $Y$ are independent continuous random variables having densities $f_X$ and $f_Y$ , respectively. What is the probability of $P[X < g(Y)]$ being $g(\cdot)$ a continous ...
4
votes
2answers
92 views

Central Limit Theorem. How to apply to the task.

The research showed that the probabilities of 3, 4, 5, 6 and 7 cars broken on one day are 0.3, 0.4, 0.2, 0.08, 0.02 respectively. If 221 car broke in 50 days, does it show that more cars break than ...
4
votes
1answer
46 views

Convergence of Random Variables in mean

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r| $$ for every $r\ge 1$ This is the very notation used. I believe it should be: $$E[|X_n|^r]\rightarrow E[|X|]^r $$ Attempt I ...
4
votes
1answer
99 views

Distribution function of the sum of poisson and uniform random variable.

Merry Christmas to everybody. I am working on the following problem. Let $X$ and $Y$ be independent Poisson($\lambda$), respectively Uniform$(0,1)$ random variables. Find the distribution function of ...
4
votes
1answer
102 views

Survivor function of a variable that has discrete and continuous components

I'm currently reading The Statistical Analysis of Failure Time Data by Kalbfleisch and Prentice and had trouble at arriving at the expression for the survivor function of a random variable $T$ having ...
4
votes
1answer
59 views

What does this hint mean and how is it useful to solve the problem?

I am doing a problem on convergence of random variable. There was a hint given, but I am still struggling to understand the hint. Here is the problem: Let $Y_n$ be uniformly distributed on ...
4
votes
1answer
231 views

Prove that it is a random variable iff it is constant on each partition

Let $\mathcal{G} = \{A_1, \ldots, A_n\}$ be a partition of a set $\Omega$, $\mathcal{F} = \sigma(\mathcal{G})$. Prove that $X : \Omega\to\mathbb{R}$ is a random variable if and only if it is constant ...
4
votes
2answers
215 views

Confidence interval of a random variable with infinite mean. (St. Petersburg paradox)

Let $X_i$ be random (independent) discrete variables such that $$\forall k\ge 0 \quad P(X_i=2^k)=2^{-(k+1)}$$ $$\begin{array}{c||ccccccc} v & 1 & 2 & 4 & 8 & 16 & 32 & ...
4
votes
1answer
121 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 ...
4
votes
2answers
208 views

Kolmogorov's maximal inequality and convergence of random series.

Let $(X_n)_{n\ge 1}$ be a sequence of mutually independent random variables, on the same probability space, with expectation 0 and finite variance. Let $S_n = \sum_{l=1}^n X_l$. Prove that for any ...
4
votes
1answer
164 views

Expected Value of a Randomly decreasing function

We are asked to find the expected value of the following function RDF(N, K) for i = 1 to K do N = random(N) return N ...
4
votes
1answer
279 views

Puzzle : Birds on a circular wire

The problem is taken from my course on randomized algorithms : There is a circle made of wire. n birds (assume n>2) occupy uniformly random position over it (visualize each bird occupying a point on ...
4
votes
2answers
144 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 ?