Questions tagged [random-variables]

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

A random variable $X: \Omega \to E$ is a measurable function from a set of possible outcomes $\Omega$ to a measurable space $E$. The technical axiomatic definition requires $\Omega$ to be a sample space of a probability triple. Usually $X$ is real-valued.

The probability that $X$ takes on a value in a measurable set $S \subseteq E$ is written as :

$$P(X \in S) = P(\{ \omega \in \Omega|X(\omega) \in S\})$$

where $P$ is the probability measure equipped with $\Omega$.

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Distribution of the first passage time of a Gaussian random walk

Does anyone know the distribution for the first passage time of a Gaussian random walk i.e. $$ S_n = \sum_{i=1}^n X_i $$ where $X_i$ are iid normally distributed random variables. The first passage time is $$ \tau = inf\{n: S_n \geq C\} $$ where $C$…
Verd
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Probability computation, tossing two dice

I have some ideas on how to solve the problem, but simulations do not support my analytical results :) Toss two dice and sum their value and write it down: Denote by $X_n$ the result at $n$-th toss. Clearly $$P(X_n = k)=\frac{k-1}{36}.$$ I would…
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Martingale and mean squared error

In preparation for a course I am doing later in the semester I have been trying to brush up on my knowledge about martingales. But I am struggling with the following problem: Let $X,Y_1,Y_2,Y_3,\ldots$ be any sequence of random variables, with…
Lech121
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Given X and Y are correlated and Y and Z are correlated what is the range of correlation between X and Z?

How can I calculate the range of correlation of two variables X and Z given I have the correlations of X and Y, and Y and Z? I've found a few resources around, namely this, but I'd like a research paper (if any). Thanks!
Travis Liew
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Is there a sequence of i.i.d. random variables that is eventually monotonically decreasing?

Here is the problem I'm struggling with: Let $(X_n)$ be is a sequence of independent and identically distributed random variables. What is the probability that the sequence is monotonically decreasing from some point on? First of all, I don't…
Leo
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Distribution over the time it takes for a random process to reach an upper threshold

I am trying to figure out a way of determining the distribution over the time it takes for an arbitrary random process to cross a threshold value. For example, a simple (solved) case would be the Poisson process. In the Poisson process, the…
rkp
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(Updated) How to compute expectation and variance of an argument of a complex random variable?

Assume that $\xi$ is a complex random variable. Its argument $\arg \xi$ is a real random variable. I am interested in how to computed expectation and variation of $\arg \xi$. Edit: I add more specifics to the question. Let $\varepsilon_j$ is a…
ashim
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Hypothesis testing: find the UMP test

Suppose $X_1,\dots,X_n$ are i.i.d. They are distributed as follows: $P(X_i > x)=(1+x)^{-\lambda}$ where $x\geq 0$ and $\lambda> 0$. I have to test the following hypothesis with level $\alpha_0$; $H_0:\lambda\leq \lambda_0$ versus…
Badshah
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Multivariate Gaussian decomposition

I've seen around the claim that an $n$-dimensional Gaussian random variable (say, having unit covariance) can be decomposed into the product of two independent random variables. $$U=ZS$$ where $Z$ is a scalar representing the distance from the…
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Function of a set of r.v.'s measurable w.r.t. the $\sigma$-algebra generated by the r.v.'s

Let $X_1, \dots, X_n$ be a set of random variables defined on a probability space $(\Omega, \mathcal{F}, P)$ and denote by $\mathcal{S} = \sigma(X_1, \dots, X_n)$ the $\sigma$-algebra generated by the random variables. Take an arbitrary function $f:…
Ivan
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How confident am I to say my random number generator is broken?

related question: Should I put number combinations like 1111111 onto my lottery ticket? It's well-established from the previous question that combinations like 111111 are not less favored in a lottery draw. So here's my new question: if I have an…
arax
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Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we wanted to find the expected value after throwing a…
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Evaluate spatial variation of density-like scalar

Apologies if this has been asked previously, but I'm not totally sure of the best way to pose the question. Background I'm evaluating the variation of a spatially varying scalar field $p$ (specifically, the density of power) across a domain. As a…
acroz
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Variance of a random variable X

Why is variance of a random variable bounded by $Var(X) \leq \mathbb{E}\left[\left(X-a\right)^2\right] $for any constant a ?
user100423
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PDF and CDF of the division of two Random variables

I have two RVs; their PDF are as the followings: \begin{split} f_{X}(x) = \frac 1 {a} e^{-\frac x {a}}\end{split} and \begin{split} f_{Y}(y) = \frac {y^{L-1}} {b^{L} \Gamma (L)} e^{-\frac y {b}}\end{split} where; a,b [Element] Reals && a,b > 0; L…
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