3
votes
1answer
30 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
66 views

“Poissonization” and intuition

In a french book, "Calcul des probabilit├ęs" 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
26 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 ...
0
votes
0answers
22 views

How large does $m$ have to be to get unique values with high probability? [duplicate]

We can suppose we are given two naturals, $r$ and $n$. We can then pick $n$ unique naturals: $\{x_0, x_1, \dots, x_n\}$. The following function is important: $$\prod_{k=1}^n{(x_k)^{y_k}} (\mod m)$$ ...
8
votes
1answer
140 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
129 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
190 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
73 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
66 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
111 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
311 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
93 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
99 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
70 views

A basic doubt regarding relative frequency and axiomatic definition of probability

I know that in the case of the axiomatic definition of probability, probability is defined as a function from the set of all events to $[0,1]$. But in the case of a relative frequency approach, can ...
0
votes
0answers
31 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
167 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
93 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
273 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 $\mu_i$ ...
3
votes
1answer
200 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
123 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
381 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) ...
2
votes
1answer
65 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
53 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
92 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
234 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
270 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
262 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
65 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
133 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
276 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
204 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
127 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 ...
4
votes
2answers
660 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
503 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
152 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
150 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
186 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 ...
12
votes
5answers
981 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
849 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
105 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
283 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 ...
8
votes
4answers
2k views

Intuitive explanation of variance and moment in Probability

While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment. What is a good way to think of those two terms?