0
votes
0answers
8 views

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 ...
2
votes
4answers
420 views

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 ...
1
vote
1answer
57 views
+50

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 ...
2
votes
3answers
68 views

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 ...
1
vote
3answers
215 views

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 ...
4
votes
2answers
66 views

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 ...
0
votes
2answers
40 views

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 ...
2
votes
0answers
44 views

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 ...
3
votes
3answers
80 views

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?
3
votes
1answer
66 views

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 ...
3
votes
1answer
46 views

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: ...
1
vote
1answer
116 views

“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 ...
0
votes
1answer
45 views

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 ...
8
votes
1answer
106 views

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. ...
8
votes
1answer
175 views

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 ...
2
votes
2answers
164 views

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 ...
0
votes
0answers
193 views

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 ...
0
votes
1answer
100 views

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 ...
0
votes
1answer
69 views

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 ...
3
votes
3answers
114 views

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. ...
8
votes
4answers
356 views

St. Petersburg Paradox

A fair coin will be tossed until a heads results. You will then be paid $2^{n-1}$ dollars where $n$ equals the number of flips. Now why is the expected pay out infinite? $$ \sum_{n \geq 1} ...
4
votes
1answer
96 views

Monty Hall vs. Card Example

In class, while illustrating the topic of conditional probability, my professor presented the following card example: You have 3 cards that have been randomly shuffled: card1, card2, and card3. ...
2
votes
1answer
123 views

Understanding the “Birthday Problem”

I found on this website http://www.cut-the-knot.org/do_you_know/coincidence.shtml proof that the probability of two people in a room having the same birthday equates to 50% when when there are 23 ...
0
votes
0answers
32 views

A basic doubt on the relative frequency approach of probability

I need to find the probability that "Ram is a boy" given that "Ram is a good boy". I understand the problem of finding this probability using the relative frequency approach. How to solve this problem ...
2
votes
3answers
197 views

Yet Another Monty Hall Question - Please advise if alternative scenario proves the same principle

Okay, I'm very embarrassed that there are already 71 questions (based on search of "monty hall") and I'm going to post another one. I read the first 5 before succumbing to choice-overload. I'll try to ...
0
votes
1answer
111 views

cauchy schwarz equality: difference in proving style for linear algebra and expectation version

I am interested in proving the following sub version of Cauchy Schawrz equality. 1) LA version : If $x$ and $y$ are two real vectors and the following holds $$<x,y> = ||x||.||y||$$ then $x$ ...
3
votes
4answers
341 views

A basic intuition on a probability problem

Two players take turns shooting at a target, with each shot by player $i$ hitting the target with probability $p_i$, $i=1,2$. Shooting ends when two consecutive shots hit the target. Let $\mu_i$ ...
4
votes
1answer
253 views

Intuition of law of iterated logarithm

Let $X_i$ be iid random variables with $EX_i = 0$ and $Var X_i=1$ and $S_n=X_1+\cdots+X_n$. Then the law of the iterated logarithm says almost everywhere we have ...
0
votes
2answers
138 views

Does $P(A\cap B) + P(A\cap B^c) = P(A)$?

Based purely on intuition, it would seem that the following statement is true, when thinking of the events as sets: $$P(A\cap B) + P(A\cap B^c) = P(A)$$ However, I am not sure if this is true, and ...
2
votes
1answer
533 views

Confusion about Banach Matchbox problem

While trying to solve Banach matchbox problem, I am getting a wrong answer. I dont understand what mistake I made. Please help me understand. The problem statement is presented below (Source:Here) ...
3
votes
1answer
73 views

Basic Question about linearity of expectation

I am going through some introductory notes on probability here http://www.stat.berkeley.edu/~aldous/134/gravner.pdf In Chapter 8, page 89, there is a problem where you get a bag containing 10 Black, ...
3
votes
1answer
58 views

Expected Value of Students on a Bus

There's a question in my probability book that says there are $148$ students on $4$ buses containing $40, 33, 25, 50$ students, respectively. If we let $X$ denote the number of students that were on ...
5
votes
2answers
106 views

Intuition for scale of the largest eigenvalue of symmetric Gaussian matrix

Let $X$ be $n \times n$ matrix whose matrix elements are independent identically distributed normal variables with zero mean and variance of $\frac{1}{2}$. Then $$ A = \frac{1}{2} \left(X + ...
4
votes
2answers
312 views

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, ...
6
votes
1answer
287 views

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 ...
2
votes
4answers
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 ...
1
vote
1answer
73 views

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 ...
8
votes
3answers
144 views

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 ...
17
votes
1answer
324 views

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 ...
1
vote
2answers
230 views

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 ...
1
vote
1answer
131 views

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 ...
5
votes
2answers
734 views

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 ...
3
votes
2answers
604 views

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 ...
5
votes
1answer
156 views

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 ...
2
votes
2answers
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 ...
2
votes
2answers
212 views

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 ...
14
votes
5answers
1k views

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 ...
3
votes
2answers
950 views

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 ...
5
votes
1answer
107 views

Diffusions - global and local

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 $\mu$ ...
3
votes
5answers
300 views

how to explain that Prob[heads, tails] = 2 * Prob[heads, heads] to a student?

I throw two coins (simultaneously). A student (very much a beginner in both math and probability theory) thought that the following 3 outcomes are equally likely: "two heads", "two tails", "a head and ...