Mean squared error of MLE of $\theta$ where $f(x) = 3 \theta e^{-\theta x^3}...
Given $f(x) = 3 \theta e^{-\theta x^3} 1_{(0,\infty)}(x)$ I want to find the MSE of the MLE estimator for $\theta$. I've found that $\hat{\theta} = \frac{n}{\sum_{i=1}^n X_i^3} = \frac{1}{\bar{X^3}}$.I...
View ArticleMean squared error of MLE of $\theta$ where $f(x) = 3x^2 \theta e^{-\theta...
Given $f(x) = 3x^2 \theta e^{-\theta x^3} 1_{(0,\infty)}(x)$ I want to find the MSE of the MLE estimator for $\theta$. I've found that $\hat{\theta} = \frac{n}{\sum_{i=1}^n X_i^3} =...
View ArticleFind the expected value of the sum of squares of the residuals.
Question: In $\mathbb{R}^2$, it is given $n$ points $(x_i,Y_i)$, where $x_i$ are known constants, and given independent random variables $Y_i\sim U(l_i,r_i)$. Fit a straight line through these $n$...
View ArticleProof of E(X * E(X|Y)) [closed]
I'm seeking a proof for the expression E(X * E(X|Y)), where X and Y are random variables. I'm particularly interested in understanding how this expression is derived rather than just its...
View ArticleAnalytical solution for minimizing a functional involving a double integral
The core of diffusion models in machine learning is to find a denoiser function to approximate the diffusion noise. However, under simple settings, I would like to know if the optimal "denoiser" $f$ is...
View ArticleIntegral with two Exponential Distributions
X~exp($\lambda$) and Y~ exp($\mu$) are two independent variables.I am trying to figure out how to solve for $E[X^{\theta -1} 1_{X<Y}]$.I believe this could be expressed...
View ArticleFormula for expectation as an integral of probability [duplicate]
I am trying to prove that for a random variable $X$,$$E(X)=\int_{[0,\infty)}P(X\geq x)dx$$I thought about first proving it for simple functions and then using monotone convergence to get nonnegative...
View ArticleExpected Maximum Value of 10 Randomly Selected Balls from an Urn
There are $20$ balls in an urn labeled from $1$ to $20$.You randomly pick $10$ balls out of this urn. What is the expected maximum value of the $10$ balls you picked out?I was able to solve the problem...
View ArticleExpected value of removing k elements and computing the penalty [closed]
Let's imagine the following situation, you have $n$ numbers from $1$ to $n$ in a row and you've removed one element from the sequence, chosen equiprobably, $k$ times. Now let's define the penalty, to...
View ArticleCalculate expected value of a function of IID samples
Suppose $X$ is a random variable and $E[X] = \mu$. We define random variable $T$ that for every IID sample $S = \{x_1,..., x_n\}$, then $T(S) = \frac{1 + \sum\limits_1^n x_i}{n}$.Although it is obvious...
View ArticleExpected number of rolls until all dice are removed (followup)
Suppose we roll $n$ dice, remove all the dice that come up 1, and roll the rest again. If we repeat this process, eventually all the dice will be eliminated. How many rolls, on average, will we make?I...
View ArticleRelated to Laplace Transform and Expectation operator
in my research work I came across the following expression:$P_1 = \mathbb{E}_{I}\bigl[\exp(-\tau\cdot f\cdot I)\bigr]$---(1)where $f,\tau$ are constant and $I$ is random variable.$P_2 =...
View ArticleEfficient way of choosing lottery ticket numbers
You want to buy a given number $n$ of lottery tickets, in each of which you have to guess exactly 6 numbers from 1 to 100. If you get 0 or 1 numbers right, you don't get any money. If you get $i>1$...
View ArticleExpected number of collsions in hashing
Suppose we use a hash function h to hash n keys into m slots. Assuming simple uniform hashing, what is the expected number of collisions? (CLRS, 3rd edition, problem 11.2-1)My solution is as...
View ArticleA problem of probability which is related to recursion in a game
Recently I have been playing a game called Warframe. I found a curious problem of probability for me in the game. I don't know how to calculate it, so I want to ask for help.The problem could be...
View ArticleAn upperbound for the variance of mean estimator in higher dimensions
I am trying to learn from the paper [paper] http://proceedings.mlr.press/v139/karimireddy21a/karimireddy21a.pdf[proofs] http://proceedings.mlr.press/v139/karimireddy21a/karimireddy21a-supp.pdf ,where...
View ArticleShow that the generating function $g_Y$ of $Y$ is given by $g_Y(z) =...
Given: Let $X$ and $Y$ be two discrete random variables with values in $\mathbb{N}_0$, such that $P(Y = k) = \frac{P(X > k)}{E[X]}$ for all $k \in \mathbb{N}$.Show that the generating function $g_Y$...
View ArticleExpected and Variance of the Squared Sample Correlation
I would like to obtain the expectation and variance of the squared sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a bivariate...
View ArticleExpectation of $\mathbb{E}[\sum \delta_i / \sum T_i ]$
Let $T = \min\{X,C\}$, where $X \sim \mathsf{Exp}(\lambda)$ and $C \sim \mathsf{Exp}(\theta)$, $X$ and $C$ independent. Let $\delta$ be a random variable such that $\delta = 1$ if $X \leq C$ and...
View ArticleWhat 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...
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