Mathematics Stack Exchange
Mathematics Stack Exchange

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 $\Omega$.

12192 questions
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Random variables related through nonlinear system of equations

Lets assume two groups of random variables X and Y (the dimensionality of them is not important). I know probability distribution of X, but not of Y. I also know that Y is a function of X and they are related through system of nonlinear equations…
Tomas
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categorical random variables formally exist?

I have a simple question: categorical random variables formally exist? Following the Wikipedia's definition of a random variable (and also in other sites), it seems that random variables are mapping functions that map real-word events always to…
Javier
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Prove that $\displaystyle \lim_{n \to \infty}\mathbb E(e^{iuX_n})=\lim_{n \to\infty}\mathbb E(e^{iuX})$ where $X$ is the mean square limit of $X_n$

Suppose $\{X_n, n \in \mathbb{N}\}$ is a sequence of random variables with finite second moment, and $X_n$ convergences in mean square to $X$, i.e., $\displaystyle \lim_{n \rightarrow \infty} \mathbb E(|X_n - X|^2)=0$, then prove…
Mengfan Ma
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Differences in random variables and estimating second moments

Suppose $S=\{{X_1,X_2,...,X_n}\} $ is a set of random variables with finite moments. If we know the first moment of all of these variables, and also we have many observations on $\{{X_1-X_2,X_1-X_3,X_1-X_4,...,X_1-X_n}\} $ do we have anything to say…
Dr. Kandy Junior
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Range of Random Variable

I understand that the Random variable is a function such as, $\text{RV}:\sigma-\text{field}(\Bbb C) \rightarrow \text{range of random variable} \in\Bbb R$ where $\Bbb C$ denotes sample space. Is there any particular name for the range of random…
Beverlie
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Generate random points within N-dimensional ellipsoid

I'd like to generate uniformly distributed points within an N-dimensional ellipsoid, where the ellipsoid axes are randomly oriented with respect to the Cartesian axes, and the means along the ellipsoid axes do not necessarily lie on the Cartesian…
Andrew Watson
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Transformation of uniformly distributed random number

Transform random numbers that are uniform on interval [0,1] into random numbers that are uniform on the interval [-11,17] I really don't know how to start I use lets F(x) = X <= x = x
user43324
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Multiplying a random variable with itself

Let $x$ in $\mathbb{R}$, I want to compute $x^²$, but I only have a random variable $X$ such that with very high probability, say $1-(1/C)$ for $C > 0$, $|X-x| \leq \epsilon$ holds. Can anything be said about the value of $X \cdot X$?
GregS
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What is the mean value of $\max\limits_{j}X_j$?

Let $X_j$ be a random variable that is $1$ with probability $x_j^*$, and $0$ with probability $1-x_j^*$. The random variables $X_j$ are independent and $j$ belongs to $\{1,\ldots,n\}$ for some positive integer $n$. I would like to calculate the mean…
Kaalouss
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How are two random variables equal to one another?

If $X$ and $Y$ are two random variables with the exact same distribution, do we then say $$X = Y?$$ Or do we say $$X = Y \ \text{almost everywhere}?$$ And if so, why? $X$ and $Y$ are maps. Why are tiwo maps equal just because they happen to have the…
Simp
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what is the expected value of $x^TAx$?

Assume $x\in \mathbb{R}^N$ is a random variable vector (like a noise sequence). You now want to calculate the following term: $E\{x^{T}Ax\}$, where $A$ is a constant matrix. How can this expression rewritten in terms of, for example, $E\{x^Tx\}$?
ehsank
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Find mass function with 3 dice and 3 different Xs

There are $3$ dice you roll one at a time, $X$ is the number of distinct numbers, as in, $X=1$, you have $(1, 1, 1)$ since there is $1$ distinct # $X=2$, $(1, 2, 1)$ or $(2, 1 ,1)$ etc... $X=3$ all different as in $(1, 2, 3)$ Find the probability…
Johnson
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Limit on the variance of a positive random number

Let's say a friend tells me he needs my help for chucking wood. He tells me that this takes 10 minutes on average. This motivates my following question. Given an expectation value E on a positive random variable and assuming that it's variance…
physicsGuy
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Independence of random sum variables

Let $(T_i)_{i \in \mathbb{N}}$ be a family of i.i.d. random variables where every $T_i \sim\mathrm{Exp}(\lambda)$. Now let $$Y :=\sum\limits_{j=1}^N T_j$$ such that for all $1 \leq j \leq N-1$ we have $T_j < c$, and $T_N \geq c$. In other words, we…
G. Bach
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Question about $L^1$ convergence for random variables

For a random variable $X \colon \Omega \to \mathbb{R}$ and a sequence of random variables $X_n$ with $$ \lim_{n \to \infty} \mathbb{E} [|X_n -X|] = 0,$$ I have found that $$ \lim_{n\to \infty} \mathbb{E} [f \circ X_n] = \mathbb{E}[f \circ X] \quad…
Adam
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