Condition for $E[X\log^+(X)]$ to be finite for a random variable $X$ [closed]
I’m trying to prove the following statement, for a random variable $X$ on some probability space:$$\mathbb{E}[X\log^+(X)] \lt \infty\quad \Longleftrightarrow \quad \int_1^\infty \int_1^\infty...
View ArticleIs my extension of the expected value on bounded sets unique?
Let $n\in\mathbb{N}$. Suppose $A\subseteq\mathbb{R}^{n}$ is Borel and $\mathcal{U}$ is the set of all unbounded $A$. Also, $\text{dim}_{\text{H}}(\cdot)$ is the Hausdorff dimension while...
View ArticleIs there a unique way to define 'satisfying' of a satisfying extension on the...
Let $n\in\mathbb{N}$. Suppose $A\subseteq\mathbb{R}^{n}$ is Borel and $\mathcal{U}$ is the set of all unbounded $A$. Also, $\text{dim}_{\text{H}}(\cdot)$ is the Hausdorff dimension while...
View ArticleExpected frequency of a letter in a sampled word with replacements
Consider the setting where we have the following:An alphabet of $M$ letters.Each letter $j$ in this alphabet has a probability $p_j$ of being sampled, and $\sum p_j = 1.We sample letters with...
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Some questions on condtional expectation-Is this a sigma algerba, Is $X$...
Q1 Is this $\mathscr{G}$ a sigma-algebra? I don't think so because $\{a,b,c\}\notin\mathscr{G}$.Q2 If it is not sigma-algebra, How is that possible to condition on non-sigma algebra? Most of relevant...
View Articlelinear regression, expectation and mean squared error
Let us assume that data is generated according to a true model $$y_i = \beta_{true}x_i + \epsilon_i$$for $i = 1, ..., n$Assume that $x_i$ are fixed, and $\epsilon_i$~ N(0, $\sigma^2$) independently.Let...
View Articleinterpret a triplet of real random variables [closed]
Let (X, Y, θ) be a triplet of real random variables, where X and Y are observed and θ is not. Moreover assume:θ∼ N (0, σθ2)X | θ∼ N (αθ, σX2)Y | X, θ∼ N (βθ , σY2) =>where α, β, σθ2, σX and σY are...
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 ArticleHow to calculate the trace of this matrix?
How do we calculate the trace of $(\mathbb{I}-X(X^\prime X)^{-1}X^\prime)\mathbb{J}_n$? This question stemmed from the below problem I came across:Suppose we have the linear regression model:...
View ArticleUsing differentiation under the integral sign for computing the gradient of...
I am following the work from Kingma and Welling, where they introduce Variational Autoencoders. To train such models they use the so called Evidence Lower Bound and maximize it. One way to solve this...
View ArticleThe expected number of moves required for a simple reflex agent to move squares.
I am reading a book called "Artificial Intelligence: A Modern Approach". The following sentence appears:"It is easy to show that the agent will reach the other square in an average of two steps."To...
View ArticleComputing expectation under change of random variable
I am following the work from Kingma and Welling, where they introduce Variational Autoencoders. To train such models they use the so called Evidence Lower Bound and maximize it. One way to solve this...
View ArticleEquivalence of finite expectation
Here is the question I am working on:Suppose $X_1,X_2,\dots$ are i.i.d. random variables. Prove that the following are equivalent:(a) $\frac{X_n}{n} \rightarrow 0$ a.s.(b) $\mathbb{E}|X_1| <...
View ArticleExpectation of the product between two dependent Bernoulli random variables
I have two dependent Bernoulli random variables $X$ and $Y$, and I know that:$$\begin{align}P(X=0)&=P(X=1)=1/2 \\P(Y=1)&=11/24 \\P(Y=0|X=0)&=1/3 \\P(Y=1|X=1)&=1/4 \\\end{align}$$Is it...
View ArticleFind the expectation of the random variable $X^3$ when $X$ is a uniform...
Find the expectation of the random variable $X^3$ when $X$ is a uniform random variable in $(0,1).$I tried solving the problem as follows:$X$ is a uniform random variable (URV) implies that, if $f_X$...
View ArticleBasketball Betting Pool - Expected Number of Points...
There are 64 teams who play single elimination tournament, hence 6 rounds, and you have to predict all the winners in all 63 games. Your score is then computed as follows: 32 points for correctly...
View ArticleExpected value of a time series
I'm reading the fourth edition of Shumway and Stoffer's "Time Series Analysis and Its Applications" and I got stuck trying to determine the expected value or mean function of a simple time series using...
View ArticleExtending Quicksort's $\mathcal{O}(n \lg n)$ Bound to Duplicated Elements
This is the final part of Problem 7-2 of CLRS'Introduction to Algorithms. The exercise asks to modify the argument given in the text so that the $\mathcal{O}(n \lg n)$ bound also applies to arrays $A$...
View ArticleLimit of Expectation values involving exponential i.i.d random variables
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent and identically distributed (i.i.d) random variables with $\mathbb{E}[X_1] = 1$ and $\mathbb{V}[X_1] = 1$. Show that$$\lim_{n \rightarrow...
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$\mathbb{E}(\varphi(X, Y) | \mathcal{G}) = \mathbb{E}(\varphi(X, Y))$ What's...
If there isn't a name for this like Doob-Dynkin-Brown-Markov Tower Lemma / Theorem, then at least what's going on here so that I can describe this proposition in words?(I guess the ff is in probability...
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