All Questions
Tagged with geometric-probability statistics
29
questions
2
votes
2
answers
140
views
Distribution of a combination of four uniformly distributed variables: $ X_1+X_2 +\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}$
My problem involves four random variables $X_1, Y_1, X_2, Y_2 \sim U(0,1)$ in the expression $Z = X_1 + X_2 + \sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}$. From what I understand so far, I need to find the ...
- 23
0
votes
0
answers
51
views
Conditional probability, two vector valued RVs on a circle.
Given two vector valued RVs $ X,Y \in S_1$ (the unit circle) that are uniformly distributed. We choose a fixed vector $a \in S_1$, e.g. in x-direction: $a=e_x$.
We then sample once from both ...
- 1
1
vote
0
answers
79
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Draw a handful of random vectors iid. Is projecting one onto the handful essentially the same as projecting one onto another?
Fix some distribution $D$ over the unit sphere in $\mathbb{C}^n$.
For $k<n$ and $x_0,x_1,\dots,x_k \overset{iid}\sim D$, call $X=[x_1,\dots,x_k]$ and identify the projection onto $\operatorname{...
- 4,886
1
vote
1
answer
48
views
Probability that the average of a binary sequence deviates from $\frac{1}{2}$.
Is there a known estimate in terms of $n\in\mathbb{N},\varepsilon>0$ of the probability that a random sequence $x_1,\dots,x_n$ with $x_i\in\{0,1\}$ satisfies that for any $k$ with $2k<n$ we have ...
- 3,545
0
votes
1
answer
123
views
Probability of a triangle inside a square
Question
If we have the square with vertices at the $4$ corners of $(0,1)^2$, and we choose a random point $z$ inside the square, the triangle is between $(0,0)$, $(1,0)$ and $z$, what is the CDF and ...
- 77
2
votes
0
answers
93
views
Is the induced Fisher information metric equal to the Fisher metric of a submanifold?
If $M = \{p_\theta : \theta \in \Theta \subset \mathbb R^d \}$ is a statistical manifold parametrized by $\theta$, with the Fisher information metric
\begin{equation*}
g_{ij}(\theta) = \int_{\mathcal ...
- 31
2
votes
1
answer
80
views
Combining probability density funciton and probability mass function
I was working on this problem:
...
- 161
1
vote
2
answers
65
views
Prove that if $X \sim$ Geometric $(p)$ then, $E(X)=\frac{q}{p} \quad \operatorname{Var}(X)=\frac{q}{p^{2}} \quad m_{X}(t)=p\left(1-q e^{t}\right)$
Prove that if $X \sim$ Geometric $(p)$ then,
$E(X)=\frac{q}{p} \quad \operatorname{Var}(X)=\frac{q}{p^{2}} \quad m_{X}(t)=p\left(1-q e^{t}\right)$
My work:
$$
\begin{aligned}
E(X) &=(0)(p)+(1)(q p)...
1
vote
0
answers
169
views
Calculating probability: Buffon-Laplace Needle Problem
Question: Find the probability $P(l,a,b)$ that a needle of length $l$ will land on at least one line, given a floor with a grid of equally spaced parallel lines distances $a$ and $b$ apart, with $l<...
- 11
0
votes
1
answer
490
views
Which probability formula do I use to solve the question?
An oil prospector will drill a succession of holes in a given area to find a productive well. The probability he is successful on a given trial is 0.2.
What is the probability that the tenth hole ...
- 533
1
vote
1
answer
402
views
Marginal Distributions obtained by restricting a 2D Gaussian to a circle
Suppose I have a 2D Gaussian
$$
f(x, y) = \frac{1}{2\pi\,\sqrt{\text{det}(\Sigma)}}\exp\left\{-\frac{1}{2}(\boldsymbol{x}- \boldsymbol{\mu})^\top \Sigma^{-1} (\boldsymbol{x}- \boldsymbol{\mu})\right\} ...
- 5,247
0
votes
2
answers
324
views
A, B, and C Roll Dice
Hello I have the following question:
Three players A, B, & C take turns rolling a pair of dice. The winner is the first player who obtains the sum of 7 (P[7] = 1/6) on a given roll of the dice. If ...
- 35
2
votes
0
answers
65
views
consolidate codependent continuous probability distributions into one multivariable distribution: how?
Sorry if this has been answered satisfactorily elsewhere; if it has and eluded me, post the link and I shall close this.
This post is motivated by a couple of questions that I’ve asked recently (at ...
user867884
3
votes
2
answers
84
views
Closed-form formula for $E_{x}[\max(u^\top x,0)\max(v^\top x ,0)]$ where $u,v$ are fixed vectors in $\mathbb R^d$ and $x$ is uniform on the sphere
Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $u,v$ be fixed nonparallel vectors in $\mathbb R^d$
Question. Is there a closed-form formula for $f(u,v) := \mathbb E_{x}[\...
- 9,575
2
votes
1
answer
137
views
If $X$ is a nonnegative $\sigma$-subGaussian random variable with $P(X=0)\ge p$, what is a good upper bound for $P(X \ge h)$?
Let $X$ be a nonnegative random variable and let $\sigma \in [0,\infty)$ and $p \in (0,1)$ such that
(1) $P(X=0) \ge p$
(2) $Var(X) \le \sigma^2$
For $h \ge 0$, define $c_X(h):=P(X \ge h)$. The ...
- 9,575