Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
12,560 questions
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Getting a function of two variables from two functions [closed]
I'm trying to get a function of two variables
$y(x, z)$
from two other functions
$y(x) = 0,7504x^2 + 0,087x - 0,0009$
and
$y(z)= 1726 \times z^{-2}$
But I don't know how to do it.
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Connections between similar looking theorems
Consider the following three theorems:
If $m,n$ are relatively prime, then $\varphi(mn) = \varphi(m)\varphi(n)$ (Where $\varphi$ is the totient function, the Euler-phi function)
If $f: A \rightarrow ...
- 167
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Joint measurability Q-Wiener process in L^2
In the book by Da Prato on SPDEs the following is claimed: suppose W is a Q-Wiener process with values in $L^2(\mathbb{R}^d)$. Thus, by the very definition, $W_t(\cdot)$ is a random variable with ...
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Rescaled subadditivity of Wasserstein distances
My question is about section 7.4 of Villani's book Topics in Optimal Transportation.
There, proposition 7.16 (ii), on page 220, states the following.
Let $X$ be a normed space, and $p\geq 1$. ...
- 402
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Winning/maximum probabilities of shifted random variables
Let $X_1,\dots,X_n$ be independent, positive continuous random variables, and let $M_X=\max\{X_1,\dots,X_n\}$. Assume that the "winning probabilities" of $X_1,\dots,X_n$ satisfy:
\begin{...
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An upper bound for the moment generating function $f(t) = \mathbb{E}[e^{tX}]$ when the mean, variance and upper bound of $X$ are given
Suppose a real random variable $X$ has an upper bound $c > 0$ but has no lower bound. $\mathbb{E}[X] = 0$ and $\mathbb{E}[X^2] = \sigma^2$ ($\mathbb{E}[]$ denotes the expectation of random ...
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Follow-up to problem with my approach with "breaking stick at two points"
I'm trying to solve a "breaking a stick of length 1 at two points uniformly at random" problem. You are asked to find - with the same setup - the expected lengths of the shortest, middle, ...
- 513
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2
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$N$ boxes with 2 balls in each, pick a ball randomly from a nonempty box each time, what is the expectation of 2-balled boxes after picking $N$ times?
I recently have encountered the following probability problem:
Suppose there are $N$ boxes and each box contains precisely 2 balls. Each time, we pick one ball randomly from those nonempty boxes, ...
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Simulating the door-switching problem [duplicate]
I have come across this problem which was apparently very famous some years ago, in which a person is placed in front of 3 doors: one of them has a stack of gold behind it, and the other two have ...
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How to compute the spacing distribution between random 'on' sites?
Consider $n$ equally spaced switches arranged in a line, each initially off. To each switch $1\leq i \leq n$, I assign a probability $p_i$ of being turned on, independently of the others. Of each ...
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The likelihood of a noisey coin (Bernoulli variable with observation error)
Suppose we have a coin which can be in one of two states $s \in \{0, 1\}$ where $x = P(s=1)$ is the probability of "heads". We observe $n$ independent realizations of the state of the coin, ...
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Calculating the probability distribution of steps a random affine recurrence/walk takes to reach atleast a certain value
Assuming i have $n$ affine functions (of the form $f_i := x \mapsto a_i x + b_i$ where $a_i, b_i \in \mathbb{R}$). I have a random, discerte process andapply them like this $y_{i+1} = f_r(y_i)$ where $...
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Estimating the error when averaging a function of a matrix over a collection of random matrices
In short, I want to understand how to estimate the error in calculating the average of a function on a random matrix. I expected to be able to use the standard error of the sample mean, but that hasn'...
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Limit of $\mathbb{P}(X_n>n-2 \mid X_1<\cdots<X_n)$ for $X_k\sim \mathrm{Unif}[0,k]$
Let $X_1,\dots,X_n$ be independent with $X_k\sim \mathrm{Unif}[0,k]$. Define
$$
P_n:=\mathbb{P}\big(X_n>n-2\ \big|\ X_1<\cdots<X_n\big).
$$
What is $\displaystyle \lim_{n\to\infty} P_n$?
It's ...
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Covariance of signed square root of difference of normally distributed random variables
Assume that $X,Y,Z$ are independently normally distributed (with potentially different mean and variance).
Are there some "nice" formulae for
\begin{align*}
& \mathrm{Cov}\left(\mathrm{...
