# 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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### Expected time to get on bus

Suppose buses arrive at the bus stop according Poisson process with rate $\lambda$. You get on bus if it is not full. The probability that a bus is not full is $p$ and is independent of arrival time. ...
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### Can the Law of Large Numbers be appied in Number Theory?

Usually, statistics have no place in Number Theory, but the Law of Large Numbers can be an exception, since it strictly deals with infinite cases. For example, if one throws a dice a finite number of ...
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### Looking for help with an actual solution to a stepping-stone probability problem

Back in college, I remember a professor giving us this question: In front of you is an endless line of stepping stones. You flip a coin: on heads, you step forward one stone (onto Stone 1), and on ...
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### Distribution of a $M$-dimensional random variable that is binomially sampled from another $N$-dimensional random variable, when $M \leqslant N$

Given a continuous N-dimensional random variable $X\sim P(\textbf{x}), X\in R^N$, $\textbf{x}=(x_1,x_2,x_3,...,x_N)$. For example, $X$ is a $H\times W$ Image, where $H\times W = N$. Now given the ...
1 vote
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### Triviality of the Chernoff bound for $t$ less than the expectation of the random variable

Let $X$ be a real valued random variable and $\lambda \ge 0$ . Then for any $t$, we have $P(X\ge t)\le e^{-\lambda t}\mathbb{E[e^{\lambda X}}]]$. Now we can get the best possible bound of this type ...
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### Uniform distribution of product of prime numbers modulo

Let $\{p_i\}$ denote the sequence of all prime numbers, minus a given prime $q$. Let $\{B_i\}$ be i.i.d. Bernoulli random variables. Consider then the random variable $X_n = \prod_{i=1}^n p_i^{B_i}$, ...
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### Convergence of the central limit theorem: pointwise vs uniform convergence

Let $X_1, X_2, \dots$ be a sequence of iid real random variables with finite moments (assume mean $\mu$ and variance $\sigma^2$) and call $X=\frac1N \sum_i X_i$. The central limit theorem tells us ...
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### Density given two constraints

Suppose that $a,b$ are real constants. That $X_1, X_2$ are independent real random variables with densities $\phi_1, \phi_2$ with respect to the Lebesgue measure. I know that a random variable $Y$ ...
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### Quantitative analysis Statistical test of Likert Scale to determine best option from multiple options

What is the best statistical test to be done to determine the best drawing of three drawings? Given data collected from a Likert scale on various characteristics of the drawings. Questions on ...
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### Why do singleton events imply sets in multiplicative but not in additive probability?

Let $a \geq 0$ and $0\leq b \leq 1$ and $M, N$ be two appropriate conditioning events such that, for all singletons $y = \lbrace y \rbrace$ in the sample space $Z$ and all subsets $Y$ of $Z$, the ...
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### Probability and Random Variables.

Hi, I was trying to understand this example in the book. In the first part of the question, We've to find p.d.f (probability density function). For that, we take the derivative of the given ...
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### Conditional Probability from textbook example unclear

I am reading "Mathematical Statistics with Applications", 7th edition from Wackerly, Mendelhall and Scheaffer. On example 2.23 on page 71, it is unclear how they calculate P(A|B). Example 2....
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### How to calculate Random Distribution?

I'm pretty sure this is a basic question but I can't seem to find an answer anywhere else. If I have N objects each given one of M random categories (using a simple uniform random distribution), how ...
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