# Tagged Questions

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### Conceptual question about independence and stopping times

Let $\{X_i\}_{i\in \mathbb{N}}$ be a sequence of i.i.d. random variables with common distribution function $\mu$. Consider a property $A$, such that $\mu(A)>0$. Define $T$ to be stopping time ...
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### What does it really mean when we say that the probability of something is zero? [duplicate]

Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible. But when we look at the probability from a mathematics perspective, probability is defined ...
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### counting occurence of subgraphs by counting their occurence in larger subgraphs

I have a mental block in fully understanding the following notion. Let $G$ be a graph of order $n$ and $H$ a fixed small graph of order $k \le n$. Suppose that there are $d$ copies of $H$ as an ...
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### Plausibility vs Probability

http://whatho.in/2013/plausibility-versus-probability/ refers to pp 155-156 of 533 of Thinking, Fast and Slow by Daniel Kahneman. I'll use one of Kahneman's other questions from p 156: A ...
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### Expressing the probability density function of $Ax$ in terms of the pdf of $x$

I understand that, for example, you might have a density function which measures the probability of observing an outcome in a certain interval measured in feet, but someone wishes to use meters ...
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### What is the intuition behind the Poisson distribution's function?

I'm trying to intuitively understand the Poisson distribution's probability mass function. When $X \sim \mathrm{Pois}(\lambda)$, then $P(X=k)=\frac{\lambda^k e^{-k}}{k!}$, but I don't see the ...
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### convert continuous random variable to a discrete one for the given exponential distribution

I understand that the following question requires converting continuous r.v. to discrete r.v. But How can we get a PMF from the CDF of continuous distribution? It involves dividing continuous values ...
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### General formula for dependent probability distributions

Recently I encountered the following problem: What is the mean distance between two random points on a unit square? I understand pen and paper methods for solving this exist however I'm ...
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### Why is the expected number coin tosses to get $HTH$ is $10$?

Can someone please explain why is the expected number of coin tosses to get the sequence of $HTH$ is $10$? What is the intuition and formulas behind this?
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### Kolmogorov's $0-1$ law and constant RV

Kolmogorov's $0-1$ Law: For any terminal event $A$ we have that either $\mathbb{P}(A)=1$ or $\mathbb{P}(A)=0$. Alternatively any $F_{\infty}$ measurable random variable (so basically a terminal ...
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### Motivation behind steps in proof of Hoeffding Inequality

The lemma that is proved for proving Hoeffding's inequality is: If $a\leq X\leq b$ and $E[X]=0$, $E[e^{tX}] \leq e^{\frac{t^2(b-a)^2}{8}}$ Here's a link to the proof: ...
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### “Poissonization” and intuition

In a french book, "Calcul des probabilités" from Foata and Fuchs, I found this theorem, which they call "Poissonization". "Let $(I_k)_{k \in \mathbb{N}}$ be a sequence of independent variables with ...
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### Intuition behind failure rate.

The failure rate of the exponential distribution is a constant, $\lambda$, as the exponential distribution is memoryless. So say we have that $\lambda = \frac{1}{10}$. What is that telling us? The ...
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### Deep Understanding of Independence of Probabilities

I really want to have a deep understanding of the independent probabilities of two events. That means to me that I just do not want to use and know the definition. I want to fully understand the why. ...
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### What is the intuition behind the generalized confidence interval?

What is the intuition behind the generalized confidence interval? My best description on GCI that it is the way to derive a formula to calcuate the area of the center region in a asymetry distribution ...
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### Intuition/Real-life Examples: Pairwise Independence vs (Mutual) Independence

Would someone please advance/discuss some real-life situations falsities $1, 2$? I'd like to intuit why these are false. As a neophyte, since I still need to compute the probabilities for the examples ...
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### Is it possible for the expectation of a random variable to be greater than it's range?

I am reading the following paper: http://www.cis.upenn.edu/~sanjeev/papers/focs11_sorting.pdf : and it states the following: Sample $N = 2(n+1)^2ln(n)$ points $\mathbf{x^1}, ..., \mathbf{x^N}$ from ...
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### Visualizing Markov and Chebyshev inequalities

I am helping a class on introductory probability covering Markov and Chebyshev's inequalities. I would like to give the students a nice visualization for why they are true or at least to show what ...
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### Expectation of $[Y g(X)]$

Firstly, how do I interpret $\mathbb{E}[g(X)]$. I understand $\mathbb{E}[X]$ is like the most likely outcome of a set of experiments (loosely speaking at least - not really a very maths person)? But ...
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### If $X\sim \exp(\lambda)$ and $Y\sim \exp(\mu)$ then $P(X\leq Y)=\frac{\lambda}{\lambda+\mu}$. Is there an intuitive interpretation for this fact?

I can verify this via double integrals, but I'm wondering if this can be put in the context of a Poisson process or something to give it an obvious meaning. I can't think of exactly how it would work. ...
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### Intuition Of Conditional Probability Equation

I was wondering if any one of you had any intuitive insight regarding the conditional probability equation, $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$. In my textbook, they give a mere definition, ...
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### Age of Stochasticity?

Today I came across D. Mumford's 1999 article The Dawning of the Age of Stochasticity, which is quite remarkable even after more than a decade. The title already indicates the theme, but I copy the ...
310 views

### Variance of binomial distribution

Why for $X\sim B(n,p)$ is $Var(X)=np(1-p)$? $Var(X)=\sum x_i^2 p_i -(\sum x_i p_i)^2=\sum_{r=0}^n r^2 \binom{n}{r}p^r(1-p)^{n-r}+( \sum_{r=0}^n r \binom{n}{r}p^r(1-p)^{n-r} )^2$ In my ...
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### Upcoming exam! Any good sources to learn about counting techniques and discrete probability?

If anyone has a free, online source to contribute for a certain topic/topics, please share! I'm not really looking for an intense theoretical grasp of these topics, just an intuitive understanding of ...
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### Seemingly similar but different probability games

Burger King is currently running its "family food" game in which each piece can be modeled as a scratch off game where exactly one of three slots is a winner and you may only scratch one slot as your ...
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### Understanding what $\sqrt{p}$ means for an event of probability $p$

Say I have a random event $E$ with probability $p$. There is a natural interpretation in terms of $E$ for the probability $p^2$: it's the probability that $E$ occurs twice if I perform two independent ...
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### Average run lengths for large numbers of trials: Intuition and proof

This article states that the formula for the average run lengths for large numbers of trials is:$$\frac{1}{1-Pr(event\ in\ one\ trial)}.$$ My questions What is the intuition behind this formula? Do ...
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### Problems with infinite $\Omega$, when trying to define product spaces of discrete probability spaces

Definitions In our course we defined a discrete probability space as a tuple $\left(\Omega,P\right)$, where $P:\mathcal{P}(\Omega)\rightarrow\left[0,1\right]$ and $\Omega$ is at most countable, such ...
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### The definition of independence is not intuitive

In the book "Introduction to Probability" by J. Charles M. Grinstead and Laurie Snell independent events are introduced in the following way: "It often happens that the knowledge that a certain event ...
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### Why do we need (the abstract concept of) random variables (in discrete probability models)?

What we defined: Suppose we have a (discrete) probability model $\left(\Omega,P\right)$, where $P$ is the probability function (at least, that was the way it was introduced in a course I took; that ...
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### Sorting through “algebra of random variables,” vs. “probability space,” etc

I have been reading through Wikipedia pages, and I'm still really confused. What is the difference between "algebra of random variables" and "probability space."? Are they just different words for ...
154 views

### Is it possible to determine that a coin is biased or not, by tossing it a number of times?

Is it possible to determine that a coin is biased or not, by tossing it a number of times ? I am sure that this problem has been studied,I am interested to know about the mathematics behind this ...
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### What is the reasoning behind a multinomial coefficient in a practical sense?

If you want to divide a team of 10 people into teams A, B, and C of sizes 3,5, and 2, how many divisions are possible? If you want to divide them into just teams of sizes 3,5, and 2, how many ...
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### Four men, hats and probability

I encountered the four men in hats puzzle for the first time today. My question is about a realisation I (think I) had while arriving at the solution, but I have no idea whether I've made a mistake ...
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### hypergeometric distribution problem

I am looking for some insight into a problem: Consider a group of $T$ persons, and let $a_1, a_2, ..., a_T$ denote the height of these $T$ persons. Suppose that $n$ are selected from this group at ...
Suppose $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$ is a diffusion. Is there a sense in which the dynamics are "dominated" locally by the diffusion term, and dominated globally by the drift term? If $\mu$ ...