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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].

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42 views

Circuit Probability Question from $\textit{Mathematical Statistics (7 ed.)}$

From Mathematical Statistics (7 ed.), Chapter 2, Supplementary Exercise no. 2.163: Relays used in the construction of electric circuits function properly with probability $0.9$. Assuming that the ...
2 votes
1 answer
28 views

How to derive likelihood function

I have been struggling a lot with the concept of likelihood and I'd really appreciate it if someone could verify if my understanding is correct and give input. If I understand this correcly, we pick ...
0 votes
0 answers
2 views

Sampling a vector of size $n$ which sums to $k$ sequentially

Suppose I want to sample a vector $X$ such that every entry is sampled from some distribution $p$ and all entries sum to $k$ (the entries are not independent). I thought I can simply sample $X_1$ from ...
1 vote
1 answer
45 views

How to prove $\forall p\in\left(\frac12,1\right)\quad\sum_{k=0}^\infty \left( \frac1{k+1} \binom{2k}k p^k (1-p)^k \right) \le \frac1p$?

How to prove $\forall m\ge0\quad\forall p\in\left(\frac12,1\right)\quad\sum_{k=0}^m \left( \frac1{k+1} \binom{2k}k p^k (1-p)^k \right) \le \frac1p$? Since this has to be true for all $m$ then it's ...
2 votes
0 answers
19 views

Chernoff bound of strings

Let $x\in(0,1)^*$ be a binary string. Consider m sample indices $i_1,i_2,\dots,i_m$ uniformly at random and independently of one another. Each $i_j$ is sampled uniformly at random from $[|x|]$ ...
3 votes
1 answer
71 views

Hamming distance in real analysis

Given two binary strings $x, y\in (0,1)^*$ such that $|x|=|y|$, then the set $$\delta{(x,y)}=\frac{|\{i\in[|x|]:x_i\neq y_i\}|}{|x|}$$ is called relative hamming distance. Given $x\in (0,1)^*$ and $S:$...
1 vote
0 answers
17 views

Doob's inequality for Wiener Fields

Let $W^{(n)}$ a $n-$fold Wiener field, i.e. a Gaussian separable real-valued field on $\mathbf{R}_{+}^N=\left\{t=\left(t_1, \ldots, t_N\right): t_i \geq 0\right\}$ with zero mean and covariance ...
0 votes
0 answers
38 views

Show that the riffle shuffle has second largest eigenvalue is $\frac{1}{2}$

Consider the riffle shuffle. This is known way to shuffle cards, forming a markov chain. For details, see this for example (How is a "riffle shuffle" mathematically defined?). I have seen it ...
0 votes
1 answer
38 views

How can I show that $\mathbb{E}(T)=\frac{a^2}{\sigma^2}$?

Let $(X_i)_{i=1,2,...}$ be a sequence of iid random variables such that $\mathbb{P}(X_i=\pm 1)=\frac{1}{2}$ and with $\operatorname{Var}(X_i)=\sigma^2>0$. Set $S_t=\sum_{k=1}^t X_i$ for $t=1,2,...$ ...
0 votes
0 answers
14 views

Approximating the poisson distribution using normal distribution

The number of calls $X$ to a telephone exchange during the busiest hour of the day follows a Poisson distribution $Po(λ)$. Over $8$ days, the following independent observations of $X$ have been ...
0 votes
0 answers
25 views

Request in getting the general expression.

Given a dartboard, of radius 'r', with 5 concentric circles, with numbering starting from the outermost one, from 1. The distance between each ring is 'r/5'. Assuming that the board is always hit, ...
1 vote
0 answers
39 views
+100

Smallest eigenvalue of matrix with random elements (non-central Wishart)

Suppose that $X \in \mathbb R^{d \times n}$ is a random matrix with independent entries, each of which follows the standard normal law $\mathcal N(0, 1)$, and that $M \in \mathbb R^{d \times n}$ is a ...
1 vote
0 answers
15 views

Is it Possible to Analyze Multiple Markov Chains Together?

Here is a problem I recently thought about. Suppose there are 2 Discrete Time Markov Chains: Markov Chain A (3-state chain): $$ X_t = \begin{bmatrix} 1/3 & 1/3 & 1/3 \\ 1/3 & 1/3 & ...
-1 votes
2 answers
20 views

How to find the percentage of a group that experienced a change in color.

Okay, let us give an example of what I am talking about. Let us say you have a collection of spheres, the amount doesn't matter, only percentages do. Let us say they all possess a colour, they will be ...
2 votes
0 answers
29 views

Can every $C^2$ function defined on a closed set in $\mathbb{R}^d$ be extended to $C^2(\mathbb{R}^d)$?

When reading Page 147 of the book "Continuous Martingales and Brownian Motion" by Daniel Revuz & Marc Yor, I am confused with the Remark $3^\circ$ of (3.3) Theorem (Ito's formula).In ...
8 votes
3 answers
906 views
+100

Is there a known explanation for the Feynman point?

The Feynman point is a mathematical coincidence. It states that from position 762, there are six consecutive nines in the decimal expansion of pi. Some mathematical coincidences have an explanation, ...
0 votes
1 answer
33 views

Method of Moment for Normal mixtures $p\cdot N(0, 1) + q\cdot N(\eta, 1)$

Setup Let $X_1,\ldots , X_n$ be random variables according to $$ p\cdot N(0, 1) + q\cdot N(\eta, 1),\ p\in (0, 1), q:=1-p. $$ We use method of moments to obtain the needed starting $\sqrt{n}$-...
2 votes
1 answer
1k views

probability: Relays used in the construction of electric circuits function properly with probability .9

Question: Consider circuit $B$. If we know that current is flowing, what is the probability that switches 1 and 4 are functioning properly? First I have calculated the probability of current flowing ...
1 vote
0 answers
38 views

Evaluating $\lim_{n\to\infty}\sum_{i=1}^n \exp\left(\frac{-|x-X_i|^2}{2(\sigma/n)^2}\right)$

Let $X_1,...,X_n\stackrel{iid}{\sim} \mu$ where has a density with respect to the Lebesgue measure on $\mathbb{R}$: $\mu(dx)=\rho(x)dx$. For every $x$ show$$ \lim_{n\to\infty}\sum_{i=1}^n \exp\left(\...
0 votes
0 answers
28 views

Determinism in Random Two Envelopes Paradox

The paper, “Pick the largest number”Open Problems in Communication and Computation Springer-Verlag, 1987, p152, deals with a version of the two envelopes problem where, after seeing one number, a ...
0 votes
1 answer
36 views

Probability mass function for combinatorics problem

The question is as follows: A small pond contains 15 fish: 10 blue fish and 5 red fish. A set of 5 fish are caught at random. (Once a fish is caught it is not placed back in the pond.) Let K be the ...
0 votes
1 answer
146 views

Conditional probability question: Misleading statement or lack of proper understanding?

I've been asked about the question below on the use of conditional probability. Probability questions have more often that not being a source of pain, as I tend to find many problem statements ...
7 votes
1 answer
2k views

Optimal strategy in three-player number-picking game

Consider a game in which there are three players. Call them Player $1$, Player $2$, and Player $3$. Here are the rules: Each player is supposed to select an integer between $1$ and $100$. Player $1$'...
0 votes
0 answers
15 views

Finding References for known results about mixing time of a Markov chain arising from Gaussian elimination

I am looking for references for the known results on the following problem. Let $(X_t)$ be a Markov chain on $GL_n(F_2)$, where in each step, an ordered pair $(i,j)$ is chosen uniformly at random, and ...
-1 votes
1 answer
38 views

Probabilities I'm gonna win a tournament when I'm better at one game and bad at another

I'm playing a game of chess and go. It's a three round game and to win I have to win two rounds in a row. I'm better at chess and my opponent is better at go. I can chose to play go-chess-go or chess-...
0 votes
0 answers
18 views

Stochastic Gaussian Process vs Non-Stochastic Gaussian Process?

Lately I have observed that there are two types of Gaussian Processes: Type 1: https://www.cs.toronto.edu/~rgrosse/courses/csc411_f18/slides/lec20-slides.pdf Type 2: https://en.wikipedia.org/wiki/...
2 votes
1 answer
52 views

$\mathbb E(\max(X_1,...,X_{t+1})|\mathcal{F}_t)$ where the $X_i$ are iid uniform

Let $X_1,...,X_T$ be independent and identically distributed uniform random variables on $[0,1]$. Let $$M_t:=\max\{X_1,...,X_t\},$$ $L_t=M_t-ct$ for a $c>0$ and $L_0:=-\infty$. If $\mathcal{F}_t=\...
0 votes
0 answers
9 views

Gaussian Distribution vs Gaussian Noise?

I am trying to understand Donsker's Theorem (https://en.wikipedia.org/wiki/Donsker%27s_theorem): Let $F_n$ be the \textit{empirical distribution function} of the sequence of i.i.d. random variables $...
2 votes
0 answers
29 views

Marginal distribution is normal, but the joint distribution is not. Why?

Denote the CDF of standard normal distribution as $\phi(\cdot)$. The joint CDF of $(X,Y)$ is $ F_{X,Y}(x,y) = \phi(x)\phi(y)[1-\alpha(1-\phi(x))(1-\phi(y))]$ where $\alpha \in (-1,0)\cup(0,1)$. ...
4 votes
1 answer
56 views

Expected value of stopping time of non symmetric random walk$E[\tau_{a,b}]$ is finite

Suppose we have a random walk $S_n$ that increases by $1$ with probability $p \ne \frac{1}{2}$ and decreases by $1$ with probability $1-p$. And let $a,b \in \mathbb{N}$. How can I show that the ...
-2 votes
1 answer
30 views

I'm quite unsure how to approach part (d). Would greatly appreciate any advice. Thanks in advance! [closed]

I'm unable to embed. Here's the link to the question
2 votes
0 answers
18 views

FOB Poker: Probability of a sequence of numbers (with the existence of "wild-card" number)

At work we have 2 factor authentication using a fob that generates a sequence of 6 numbers 0-9. We started playing "poker" by having everyone generate a number at the same time and see who ...
0 votes
0 answers
15 views

What is the meaning of $p_{g_i}$ in this equation?

I am reading "The element of statistical learning" and having some question regarding equation 2.36. The book stated that: "A more interesting example is the multinomial likelihood for ...
0 votes
0 answers
28 views

Why are logarithms used to measure Similarity?

I was reading this link on how to test the difference between 2 probability distributions :https://chjackson.github.io/kl.html. If $P(x)$ and $Q(x)$ are discrete distributions, the KL Divergence can ...
1 vote
3 answers
191 views

Given 3 white balls and 1 black ball in a box.Take out ball by ball until getting a black one.Calculate the expected value for number of takes

Given a container with 3 white balls and 1 black ball. We take out balls randomly from the container, one by one, until a black one pop out. Calculate the expected value of number of balls taken out, ...
0 votes
0 answers
28 views

Find a majorant of two random variables

Let $X,Y \in \mathbb{L}^{\infty}(\Omega,\mathcal{F}, \mathbb{P})$ two random variables such that $\mathbb{P}(X<0)>0$ and $\mathbb{P}(Y<0)>0$. Is it true that there exists a $Z \in \mathbb{...
2 votes
1 answer
180 views

When do these Algebra Equations have solutions?

I am learning about the Stationary Distributions and Limiting Distributions of Markov Chains (I think these equations are true for both discrete time and continuous time): $$\text{1) Stationary ...
7 votes
1 answer
396 views

What probability distribution function is this?

This is sort of a followup to this question (I'll mention everything relevant in this post though so no need to click link). Main Question: I was trying to study a random variable $Y$. I will ...
-4 votes
0 answers
53 views

I dont get how i can solve b [closed]

I tried solving it with the zentraler Grenzwertsatz and Normalverteilung but i cant get so be near the solution. Can you please help me so i understand it enter image description here enter image ...
0 votes
1 answer
34 views

Prove symmetry of probabilities given random variables are iid and have (not absolutely) continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous. Denote common distribution as $$F(y) :...
0 votes
0 answers
27 views

Asymptotic distribution involving a high-dimensional matrix

Let $ \varepsilon \sim \mathcal{N}({\bf 0}, I_n)$ be a n-dimensional multivariate normal RV and $x_1,x_2,...,x_n \sim \mathcal{N}({\bf 0}, I_p)$ be i.i.d p-dimensional multivariate normal RVs. $X$ is ...
5 votes
1 answer
952 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le \...
-1 votes
1 answer
30 views

Probability space of a random sample and almost sure convergence [closed]

When working with random samples in statistics, there are 2 approaches: To have the single-outcome space $(\Omega, F,P)$, on which $n$ i.i.d. random variables $(X_1,...,X_n)$ act. I.e., a single $\...
1 vote
2 answers
91 views

What can I say about $P(B|A_1 \cap A_2)$ if I know that $P(B|A_1), P(B|A_2) \approx 1$?

Let $(\Omega,\Sigma,P)$ be a probability space and $A_1,A_2,B \in \Sigma$. Define $P(B|A_1) = E(\chi_{B}|A_1)$. From real life, I have experience that if there are two numbers $\alpha_1,\alpha_2 \...
1 vote
1 answer
28 views

Counting squares in a regular polygon with $4n$ vertices ($n \geq 2$)

Select at random $4$ vertices from a regular polygon with $4n$ vertices ($n \geq 2$). Find $n$ providing that the probability that the $4$ chosen vertices are the vertices of a rectangle that is not a ...
0 votes
2 answers
954 views

Calculate probability that team A scores against team B

I'm trying to calculate the probability of an event for my math classes but I'm not sure if I'm doing it correct. As I'm a sports fan I'm using a football match as subject. In the last $3$ matches ...
0 votes
1 answer
43 views

Are unconditional probability and conditional probability method necessary to generate the same result?

mcs.pdf shows 2 methods to calculate the probability of "switch wins" for Monty Hall problem. unconditional probability (See Figure 17.5): So we have the probability $\frac{1}{9}\cdot 6=\...
0 votes
0 answers
72 views

Pi's digits (unlikely occurrences?) [closed]

There are no zeros in the first 31 digits of pi. There are many unlikely patternes in pi like repeating numbers, are these coincidences?
3 votes
2 answers
88 views

Want an example of uncorrelated but dependent joint Bernoulli example [closed]

Can anyone give an example (joint probability table) that two Bernoulli variables are uncorrelated but not independent?
12 votes
3 answers
3k views

How many rolls are sufficient to ensure, with probability 99%, that the sum is greater than 100?

I roll a pair of fair dice $n$ times, and calculate the sum of all $2n$ faces which come up: Suppose each roll of each die is independent of other rolls. How many rolls are sufficient to ensure, with ...

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