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$n\times n$ grid filled with $n$ colors. What is the average group size as...
Take a grid with dimensions $n\times n$ squares and randomly fill each square with $1$ of $n$ colors. What is the expected average group size of colors touching each other as $n$ approaches $\infty$?...
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Conjecture: Expected total area of a certain set of random triangles in a...
Choose $3n$ independent uniformly random points in a disk with perimeter $x^2+y^2=1$. Label the points $P_1,P_2,P_3,\dots,P_{3n}$ in order of increasing $x$-coordinates. Form triangles $\triangle...
View ArticleGrowth of the average numbers of peaks for the permutations of $n$ sticks...
There are $n$ sticks of lengths $1$ to $n$ in a row. Upon permuting them randomly, we may calculate the average number of peaks viewed from left. A peak is a stick such that all sticks to its left are...
View ArticleWhat is the expected sum of the numbers that appear on two dice, each biased...
This has been asked twice before, expected sum of the numbers that appear on two dice, each biased so that a 3 comes up twice as often as each other number?I have further questions about this regarding...
View ArticleTerminating Sequence expected length
I was preparing for a quant interview and I came across a puzzle on QuantGuide(named Sequence Terminator):A fair 6−sided die is rolled repetitively, forming a sequence of values, under the following...
View ArticleBreaking up Expectation over a Joint Distribution
For some distribution $D\in \Delta(\mathcal{X}\times \mathcal{Y} )$ and function $f : \mathcal{X}\times \mathcal{Y} \to \mathbb{R}$, is it true that:\begin{align*}\mathbb{E}_{(x,y)\sim D} [f(x,y)] =...
View ArticleSplitting 2n + 1 cards, what is the expected ratio of a size of the smaller...
Assuming equal probability of where the split is.I can find the expected size of the smaller deck of card is (1+n)/2, but I'm not sure how the find the expected ratio in a closed form of...
View ArticleSplitting $2n + 1$ cards, what is the expected ratio of a size of the smaller...
Assuming equal probability of the deck where the split is.I can find the expected size of the smaller deck of card is $\frac{1+n}{2}$, but I am not sure how the find the expected ratio in a closed form...
View ArticleExpectation of a monotone function of CDF: $\mathbb E \left [g(F(X)) \right ]$
If $X$ is a random variable with distribution function $F,$ then $\mathbb{E}[(F[X])^{-1/2}]$ can be computed by integration by parts, if $X$ has a continuous density $f$. What happens in the general...
View ArticleClosed form expression for the expected radius of a minimal circle enclosing...
First but not least, I am not interested in an algorithm for finding the smallest circle enclosing N points in the plane!Given a normal distribution of N points on a plane, centered on origo, and with...
View ArticleExpected sum of random variable can be bounded by a random variable?
Assume we have (k+1) positive non-i.i.d. random variables $(X_0, X_1,X_2,...,X_k)$ where $k$ is a constant. Moreover, we are given $E[X_i|X_{i-1}]$ is positive for any $i\geq1$. Additionally, there is...
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An interesting conjecture
This question is a very long one, so sit tight.I designed and wrote a bot (in c++) that plays blackjack. Without going into any unnecesary details, this is how it takes decisions:It takes 3 parameters...
View ArticleExpected number of coin flip to get HTT by conditioning
What is the expected number of (fair) coin flips to get a sequence HTT? I know similar questions have been asked before and that the answer should be $8$, but I can't seem to get my head around this...
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A walk on a $2D$ Poisson process in which every step goes to the nearest...
Uncle's epic journeyOne year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in our...
View ArticleExpectation of 1 / X over the reals
I'm trying to answer a question where the sample space is $\mathbb{R}$, $X(x) = \min\{2, x\}$, a random variable, $f_X(x) = 2(1 + x)^{-3}$ if $x \geq 0$ and $0$ everywhere else. Find...
View ArticleLaw of large numbers for non-independent and non-identically distributed samples
Let $X \sim p_X$ be a real-valued random variable with $\mathbb{E}[X] = \mu > c$ where $c \in \mathbb{R}.$Assume you sample from $p_X$ and only accept samples such that the current sample mean is...
View Article$\mathrm{Re}(E\{\mathbf{x}\mathbf{y}^H\} )=\mathbf {0}\Leftrightarrow...
Let $\mathbf{x}$ and $\mathbf{y}$ two complex (column) random vector of dimension $m \times 1$. Is it true that:\begin{equation}\mathrm{Re}(E\{\mathbf{x}\mathbf{y}^H\} ) = \mathbf{0}\Leftrightarrow...
View ArticleBest tactic for choosing valued items. (variant of secretary problem)
Suppose we have to fill our bag with items and each item has a value $V_i \sim Uniform[a,b]$ where each random variable is iid. There are $N$ items in total but only a fraction $x$ of all items can be...
View ArticleInvalid Conditional Probability - Expectation of Die Rolls
Problem: Roll a fair 6−sided die until a 6 appears.Given that the first 6 occurs before the first 5, find the expected number of times the die was rolled.Note: A discussion on the solution already...
View ArticleExpectation of maximum of n i.i.d random variables
I have $n$ i.i.d. random variables, $X_1,..., X_n$ which follow some arbitrary distribution. Based on experiments in Python with various distributions, it seems that $\mathbb{E}(\max(X_1,...,X_n))$ is...
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