Minimum expected number of 5-die rolls to simulate a 6-die
Context: If we only have a fair 5-sided die, we can emulate a fair 6-side die roll by rolling D5 twice. Out of 25 outcomes, we can map 24. If we get the 25th outcome, eg. (5,5), we just repeat the...
View ArticleExpected number of cards that are larger than both of their neighbors
Seven cards numbered from $1$ to $7$ are randomly shuffled and then arranged in a circle. Given that the $1$ and $7$ are adjacent to each other, what is the expected number of cards that are larger...
View ArticleExpectation of log-normal random variables
I’m working through a derivation to understand how noise in the concentration threshold $C_\theta$ affects the width 𝑤 of the transition zone in the French Flag model. The model assumes that...
View ArticleIs a random matrix over a finite field have the largest expected rank?
Does a completely randomized matrix over a finite field have larger expected rank than other imperfect random matrices? (If this statement is vague, could you clarify this point with respect to the...
View ArticleShowing bounded functions produce bounded expectation
If we have a bounded, nonnegative, measurable function $f$ which is subadditive, would it be possible to say $E[f(X+Y)] < \infty \implies E[f(X)] < \infty$?More specifically f goes from R to $[0,...
View ArticleGaussian processes for Machine Learning, Rasmussen, Exercise 5.2
ProblemConsider the difference between the log marginal likelihood given by: $\sum_i \log p(y_i|\{y_j, j<i\})$, and the LOO-CV using log probability which is given by $\sum_i \log p(y_i| \{y_j,...
View ArticleMinimum expected number of 5-die rolls to simulate a fair 6-die
Context: If we only have a fair 5-sided die, we can emulate a fair 6-side die roll by rolling D5 twice. Out of 25 outcomes, we can map 24. If we get the 25th outcome, eg. (5,5), we just repeat the...
View ArticleOptimizing Polling Times to Minimize Average Response Time Under Constraints
Background:I am working on optimizing an RPC (Remote Procedure Call) server that handles a high volume of requests (around 200,000 QPS). Each incoming request follows this procedure:Cache Check: If an...
View ArticleClarification regarding betting game and linearity of expectation
Here is the problem I'm dealing with: You have 3 blue and 3 red cards. These cards are mixed and placed face-down in a deck, ready to be turned over one-by-one. Before each card is turned, you are...
View ArticleExpected payout in a 2 player die game
You and your opponent throw a 6-sided die. The person with a larger number wins $1. If the die values are equal, neither of you wins the reward. Now, say you and your opponent have the option to pay 25...
View ArticleExpectation and Conditional Probability
An urn contains $a$ white and $b$ black balls, where $a$ and $b$ are positive integers. One ball at a time is randomly drawn until the first white ball is drawn. find the expected number of black balls...
View ArticleExpectation of $X/(X+1)$ where $X$ is a log-normal distribution
I am wondering if there is a closed-form solution for $E \left[\dfrac{X}{X+1}\right]$, where $X \sim \operatorname{LogNormal}(\mu, \sigma^2)$.Thanks.
View ArticleExpected Number of Lattice Points: Volume Based Proof
A cube of side length $\frac{7}{3}$ is dropped into a random place on the 3Dlattice in the coordinate space (a coordinate plane with an extradimension), with the sides of the square...
View ArticleDice roll, expected sum of rolls until we get $3$?
We keep throwing a fair $6$-sided dice until we see $3$. What is the expected sum of all the dice rolls, including the last one? Note: please include the last roll in the sum as well.The answer is...
View ArticleExpectation of correct ordering in blind ranking
Alice and Bob are playing a game where Bob will give Alice $4$ natural numbers, one by one, all less than $100$. Alice has to put them in an order. Alice's score will be the number of correct ordering...
View ArticleWhy is $E[f(X)|X]=f(X)?$
I am taking "random signals and noise" course, and the professor said that $E[f(X)|X]=f(X)$.I tried to prove it, even for the simpler case $E[X|X]=X$, but I did not succeed. I found the answer given in...
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Draw $n$ random chords in a circle. What is the expected number of sides of...
Draw $n$ random chords in a circle. Each chord connects two independent uniformly random points on the circle.Here is an example with $n=40$.In this example, the centre of the circle (shown in red) is...
View ArticleWhy does the median minimize $E(|X-c|)$?
Suppose $X$ is a real-valued random variable and let $P_X$ denote the distribution of $X$. Then$$E(|X-c|) = \int_\mathbb{R} |x-c| dP_X(x).$$The medians of $X$ are defined as any number $m \in...
View ArticleExpected number of steps to stop a range division game
Consider the range of positive integers $[1, N]$, which is the starting point of the game. At each step of the game, a number is chosen uniformly randomly from the current state of the array. If the...
View ArticleStrategy to roll a 10-sided die
Suppose we play a game with a 10-sided die where we roll and sum our rolls. We can stop any time, and thesum is our score. However, if our sum is ever a multiple of 10, our score is zero, and our game...
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