Questions tagged [probability]
For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].
109,313 questions
1
vote
0
answers
29
views
Probability of finding a particle with fixed angular velocity in a sphere section
This is a probability problem coming from physics originally. The big picture idea:
A particle is moving with a fixed angular velocity on a upper half-sphere (𝑧&...
- 11
2
votes
1
answer
124
views
$\operatorname{Pr}(\text{Event}) = 1$
What I know:
$0 \leq \operatorname{Pr}(\text{Event}) \leq 1$
$\operatorname{Pr}(\text{Impossible event}) = 0$
$\operatorname{Pr}(\text{Certain event}) = 1$
If now it’s raining, can I say $\...
- 945
6
votes
0
answers
89
views
Simulating the results of a differential equation
The context of my question is for (Continuous Time) Markov Chains (https://en.wikipedia.org/wiki/Continuous-time_Markov_chain).
I have a matrix $Q$ with the the properties:
Off-diagonal elements $q_{...
- 4,265
2
votes
1
answer
69
views
How to maximize the following functional?
Suppose I want to maximize:
$$\text{H}[\mathbf{x}] = -\int p(\mathbf{x}) \ln p(\mathbf{x}) \text{d}{\mathbf{x}} $$
subject to the constraints
$$\int p(\mathbf{x}) \text{d}{\mathbf{x}} = 1$$
$$\int p(\...
- 703
1
vote
1
answer
46
views
How to use the strong Markov property of Brownian motion to get the following result?
To be precised, first given a filtered probability space $\left(\Omega,\mathscr{F},(\mathscr{F}_t),P\right)$ and a $(\mathscr{F}_t)$-Brownian motion $(B_t)_{t\geq 0}$ which means that $B$ is a ...
- 169
7
votes
1
answer
102
views
Logarithmic behavior of the sum $\sum_{n=1}^N \Bigg|\frac{\sin^2(n+1)}{n+1}-\frac{\sin^2n}{n} \Bigg|$
If we consider the sum $$A(N)=\sum_{n=1}^N \Bigg|\frac{\sin^2(n+1)}{n+1}-\frac{\sin^2n}{n} \Bigg|$$
Studying the sequence we can easily derive that it have a logarithmic-like growth: let
$$
a_n=\frac{\...
- 432
3
votes
0
answers
39
views
Does uniform integrability preserved under the modification of process?
I am using the book GTM113 of Karatzas & Shereve to learn Doob--Meyer decomposition (Theorem 4.10 in Chapter 1) and found its proof confused. Here is the description of the problem.
Let $(\mathcal{...
- 141
1
vote
2
answers
77
views
Question regarding independence in set theory: Does the provided statement give independence for one set of events, or all pairs of events?
Consider set H for Health insurance, and consider sets P as well as Q for life insurance (...
1
vote
1
answer
90
views
Is $a+bi \equiv c \bmod p$ infinitely often provided that $i\equiv \sqrt{-1}$ evaluates to some value modulo a prime $p$ for all valid primes $p$?
By This Math StackExchange Question we know that the imaginary number $i$ is equivalent to some "random" integer modulo a prime $p$ only if $p$ is equivalent to $1$ modulo $4$. It is also ...
14
votes
3
answers
620
views
A problem about random permutations
The following question was asked by another user and then deleted. I liked my answer to the post, so I have made another post with the question and the answer. I am paraphrasing the question here, ...
- 495
2
votes
1
answer
69
views
Karatzas Shreve Problem 1.3.9
I am trying to solve Problem 1.3.9 in the book by Karatzas and Shreve. It considers a Law of Large Numbers for the Poisson Process and considers it as something of a corollary to the Doob inequalities....
- 23
-1
votes
1
answer
40
views
Berry-Esseen bound for U-statistics
Let us consider $X_1,\ldots X_n$ be i.i.d. random vectors taking values in $\mathbb R^d$. Let us consider the U-statistic
$$
U_n := \binom{n}{m}^{-1}\sum_{1\leq i_1<\ldots <i_m\leq n}h(X_{i_1},\...
- 137
0
votes
0
answers
42
views
Worst Case "Uniformization" of Probability Distribution
Let $S$ be a finite set and $p:S\to [0,1]$ a probability mass function on $S$, i.e. $\sum_{s\in S} p(s)=1$. For any subset $U\subset S$ define a probability mass function $q_U:S\to [0,1]$ via $$q_U(s) ...
- 668
4
votes
2
answers
197
views
Probability that 1 is a double root of a random degree-5 integer polynomial with bounded nonzero coefficients
Edited: Correct formula for $\text{Var}(S_2)$ and $\text{Cov}(S_1,S_2)$
Note: My error (not corrected below) with $\text{Var}(S_2)$ noted by btilly below due to incorrect formula I used. The correct ...
- 185
0
votes
2
answers
58
views
Showing that the covariance matrix between these two variables is diagonal.
Consider two random variables $y_1$ and $y_2$, where $y_1$ is symmetrically distributed around $0$ and $y_2 = y_1 ^2$.
$$E_{y_1,y_2}[y_1 y_2] = \int \int y_1 y_2 p( y_1,y_2) dy_1 dy_2$$
$$= \int \...
- 703
