# 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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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### 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 $[|x|]$ ...
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### 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
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### 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 ...
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### 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 ...
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### 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 $t=1,2,...$ ...
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### 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 ...
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### 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
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### 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
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $1$'...
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### 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 ...
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### 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-...
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### 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/...
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### 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 $\alpha \in (-1,0)\cup(0,1)$. ...
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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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### 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
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### 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, ...
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1 vote
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### 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 ...
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 ...