Prove that if 101 items are distributed among 10 boxes, then at least one of...
My solution$$\text{Let } N_i =\begin{cases}1 & \text{if box } i \text{ contains more than 10 items after all 101 items are distributed}\\0 & \text{otherwise}.\end{cases} \\$$Then,$$ N =...
View ArticleOptional Stopping Theorem for expected number of flips
What is the expected number of flips of a fair coin needed to have $10$ more heads flipped than tails or $8$ more tails flipped than heads ?I wanted to do this formally with the following reasoning,...
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Birthday Probability Problem with Groups (Cambridge Exam Question)
How to solve this problem? I am thinking is the expectation to be nC2 * 1/365 but and then do not have ideas. Thanks so much!
View ArticleExpected value of a recursive process
The grass in my garden grows by 1cm every week. Every Sunday morning I cut the grass by selecting half of the blades independently at random and cutting them to zero length. Assuming I have been doing...
View ArticleOptimal Strategy for a betting game.
I shall present my question and then add some of the work I have tried, unfortunately I haven't had much luck so far.The gameThe game is played between $3$ players. You and $2$ opponents who we shall...
View ArticleFind expected value of the time between a markov chain leaving x and coming...
Hey I'm having trouble in the subject of Markov chains and expected values, would've liked some clarification.Assuming we have a hyperconnected Markov chain with a transformation matrix P and a...
View ArticleExpected value of deck of cards going back to original order in top-to-random...
I have a problem in stochastic processes I would like help with:Given a 52-card deck we shuffle the cards using Top-to-random shuffling: in each turn we take only the top card and put it somewhere in...
View ArticleOptimal Strategy for Multi Player Coin Game
You are playing a game involving yourself and 4 other players, each one of you starting with \$1000. You have an unfair coin that lands heads $\frac{2}{3}$ of the time and tails $\frac{1}{3}$ of the...
View ArticleThe relationship among $\frac{1}{8}$, $\frac{(1+a)^2}{8}$, and...
What is the relationship among $\frac{1}{8}$, $\frac{(1+a)^2}{8}$, and $\frac{(1-a)^2}{8}$?
View ArticleCompute $\mathbb E[\min(X,Y)]$ [closed]
$X$ and $Y$ are both independent geometric random variables with respective parameters $p = 1/2$ and $q = 1/3$.The expected value of $\min(X,Y)$ is 1,5.
View ArticleConvergence in probability implies convergence in distribution. Proof...
I have some doubts about my proof, I would be grateful if you can point any mistakes.Statement:Let $\{X_n\}$ converge to $X$ in probability. Show that $\{X_n\}$ converges to $X$ in...
View ArticleIs $\int XdP $ the same as $\int Xd\mu $. Question about expectation $EX$...
Let $\mu$ be the distribution function of $X$. Are $\int_{\Omega} XdP $ and $\int_{\Omega} Xd\mu $ the same thing? If not how do they differ? Please elaborate. Also what exactly is that $P$ in...
View ArticleParticle in Sphere
The position for a particle is random, with uniform distribution on the sphere whose centre is the origin and radius is 7. Calculate the expected value of the particle's distance from the origin.For...
View ArticleClarification: Definition of moment / expected value - Measure Theory
Given a measure space / probability space $(\Omega, \mathcal{F}, P)$ - one might think about $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$, where $\lambda$ denotes the Lebesgue measure on...
View ArticleExpected maximum of a multivariable function from sampling
Say we have a function $f(x_1,x_2,x_3,\ldots,x_n): \mathbb R^n \mapsto \mathbb R$ and $n$ independent random variables $X_1, X_2, X_3, \ldots, X_n \in \mathbb R$.Given a positive number $s \in \mathbb...
View ArticleProbability vs Frequency
By observed freuency, one hits the target 90 times out of 100 times so we then conclude P(hit) = 0.9. Then, the probi one can hit 90 times out of 100 times = 100C90 • 0.9^90 • 0.1^10 = 0.1319, which is...
View ArticleCalculating the expected difference in observations
Let $X_1,X_2$ be i.i.d $N(0,\sigma^2)$. My tasks is to calculate the expected value of the function $T=|X_1-X_2|$. Intuitively, I want to say that this value would equal the standard deviation, but I...
View Article$E((\ln X)^m)$ for Weibull Distribution.
I want to find the $E\left[ \left( \ln\left(X\right)\right)^m \right]$ for Weibull Distribution, which is $f_X(x;\tau,\theta)=\frac{\tau}{x}\left( \frac{x}{\theta}\right)^{\tau} e^{ -\left(...
View ArticleExpected Breakup Length
The question - A stick is broken into 3 pieces, by randomly choosing two points along its unit length, and cutting it. What is the expected length of the middle part?So the answer using symmetry is...
View ArticleExpected Size of the group of guests
This question is from Quantguide (Party Groups 2):There are50 guests at a party and they are making groups for a game. To do this they each write their name on a piece of paper and put it into a hat....
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