What is the expectation of the Dirac delta function of a random variable
I have a Dirac delta function as follows:$\delta_{\epsilon = y}$ where $\epsilon \sim N(0, \sigma^2)$, and $y \in \mathbb{R}$. I want to take the expectation of this function, but it appears that it...
View ArticleExpectation of the geometric distribution
Find the expectation of a Geometric distribution using $\mathbb{E}(X)= \sum_{k=1}^\infty P(X \ge k)$. Okay I know how to find the expectation using the definition of the geometric distribution...
View ArticleExpectation and variance of the geometric distribution
How can one use memorylessness and the law of total expectation and the law of total variance to find the expectation and variance of the geometric distribution?I will post my own answer, but as...
View ArticleHow to compute the expectation and variance of a geometrically distributed...
Straightforward using definition and using MGF $$\frac{1}{(1-z)^{2}}$$i have tried using the expectation of geometric d. E(x)=x*(N1 C x)(N2 C n-x) / (N C n) but i don't think this is the the equation...
View ArticleLet $X_i\sim\text{Ber}(\frac{1}{2})$ be i.i.d. and $Y_i=\max(X_i,X_{i+1})$...
Problem: Let $X_1,X_2,\dots$ be independent random variables with $X_i\sim\text{Ber}(\frac{1}{2})$. Let $Y_i=\max(X_i,X_{i+1})$ and $Z_n=\sum_{i=1}^{n}Y_i$. Find $\text{E}(Z_n)$ and...
View ArticleExpected Value of Loss of Two independent Exponential Random Varirables
The question is the following:Losses follow an exponential distribution with mean 1. Two independent losses areobserved. Calculate the expected value of the smaller loss.My thought process was the...
View ArticleExpected number of spins to land on two distinct regions
A spinner has three regions, with respective probabilities $\frac 14$,$\frac 14$,$\frac 12$. What is the expected number of spins needed to land on two distinct regions ?My solution:Label the regions...
View ArticleExpected Value for a Game with Multiple Rounds
I was wondering what the general approach is for calculating the expected value of a game that has multiple rounds?For example, say in my game that I first flip a fair coin. If it's heads, then I get...
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 ArticleExpected number of pairs in a hand
Consider a 10-cards deck: two cards have face value 1, two cards have face value 2,..., two cards have face value 5. You pick 4 cards without replacement from this deck. What is the expected number of...
View ArticleAveraging a nowhere continuous function defined on the rationals?
Suppose, we define function $f:\mathbb{Q}\to\mathbb{R}$, where:\begin{equation}f(x)=\begin{cases}1 & x\in\left\{\frac{2p+1}{2q}:p,q\in\mathbb{Z},q\neq0\right\}\\0 &...
View ArticleHow do we prove "almost all" Borel $f$ have different expected values from...
Let $n\in\mathbb{N}$ and suppose function $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$, where $A$ and $f$ are Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, where...
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Limit of expected number of rolls before going out
Let $n \geq 2$ be a natural number and consider the following game. There is a line of $n+1$ cells and the player starts from the first cell. A fair die with $n$ faces is rolled. The player moves that...
View ArticleExpected number of subtree removal in a tree.
I was solving this problem. In a gist the problem is as follows:You are given a rooted tree. On each step you choose a node randomly and remove the subtree rooted by that node and the node itself,...
View ArticleExpected value $\mathbb{E}(\ln(\Phi(x)))$ [closed]
Suppose $X \sim \mathcal{N}(\mu,\sigma^2)$ (may not be standard one). If there's no $\ln$ in the expectation, one can try the method in this link to compute. Then we can compute $\mathbb{E}[\Phi(a - X)...
View ArticleDerivative of expectation with gradient flow $\frac{d}{dt} E[\phi(x)]$ where...
I haven't done calc in a while and this is likely a trivial misunderstanding, but I encountered this derivation of the Wasserstein Gradient Flow and I was unsure about a couple of steps.We have $x'(t)...
View ArticleExpected number of the bins that contain more than $k$ balls [closed]
Given n balls uniformly and independently distributed into $m$ bins, what is the expected number of bins that contain more than $k$ balls?$$\mathbb{E}\left(\sum_{i=1}^{m} 1_{\{n_i > k\}}\right)=...
View ArticleBalls into Bins: Expected number of the bins that contain more than $k$ balls
Given n balls uniformly and independently distributed into $m$ bins, what is the expected number of bins that contain more than $k$ balls?$$\mathbb{E}\left(\sum_{i=1}^{m} 1_{\{n_i > k\}}\right)=...
View ArticleDerivative of expectation with gradient flow $\frac{d}{dt} E[\phi(X(t))]$...
I haven't done calc in a while and this is likely a trivial misunderstanding, but I encountered this derivation of the Wasserstein Gradient Flow and I was unsure about a couple of steps.We have $X'(t)...
View ArticleSolve the following integral: $\int_{x_0}^\infty \log(x) e^{-(x-a)^2/b}dx...
I want to compute this integral preferably in closed-form without expanding the $\log$ function; however, efficiently computable approximations might also solve my problem:$\int_{x_0}^\infty \log(x)...
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