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The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.

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Explanation of proof for infinite variance in ordinary importance sampling for value function calculation

In the barto and sutton book, the authors have provided an Example on page 106, Ex 5.5 where they prove that the variance is infinite for an ordinary importance sampling method. In this derivation, ...
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Expected value of quotient of Poisson distributions

Let $X$ and $Y$ be independent random variables such that $X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$. Is there ...
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1answer
24 views

How to calculate expected risk from fitted Cox PH model in R?

I'd like to calculate expected risk (cumulative incidences), which are derived from fitted Cox PH model using R packages. I have the fitted Cox PH model like as follows: [Variables] Dataset: 10,...
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1answer
52 views

Show $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0$

Show that $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0,$ where $X, Y$ are random variables with constant means and $h(x)$ is an arbitrary function. So far, I have expanded out the expectation and used ...
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59 views

Multiple interval ratio of E[X / (X + Y)]

I have a sequence of interchanging on- and off-intervals, each pair identified by index $i$. The duration of the on-interval $i$ is represented by random variable $X_i$, and the duration of the off-...
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27 views

How do I prove in this question that E(2^X) doesn't exist?

I understand how to get E(X), but how can I derive E(2^X)?
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Batch Normalization expectation operator

On page 2 the paper Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift they pull a couple of fast moves with the expectation operator and I'm not sure why. ...
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61 views

Why is expected value of random variable equal to mean

While learning about Random variables I came across the mean of random variable X. The definition says that the expected value of random variable E(X) = Mean of Random variable X I am not able to ...
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59 views

Variance of continuous Random Variable negative?

I am having trouble finding the Variance for this question. The proportion of salt X left in the salt shakers at the end of the day at a crowded restaurant has a probability density function given ...
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25 views

Expectation of inverse non central chi-squared

If $X\sim \mathcal{N}_p(\theta,\sigma^2)$ then $X^{'}X/\sigma^2\sim$ non-central $\chi^2_p$ with non-centrality parameter $\theta^{'}\theta/\sigma^2$. How to find out the expectation of $\sigma^2/X^{...
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Comparing suprema of inner products of Gaussian variables

I'm given two i.i.d. standard normal vectors $x, y \sim \mathcal{N}(0, I_n)$, and vectors $a \in \mathbb{S}^{n-1}$, the unit sphere in $n$ dimensions. Additionally, given a set $S \subseteq [n]$, I ...
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Expectation of the minimum of dependent random variables

How do we compute the expectation of the minimum of dependent random variables? In other words, what is the value of $\mathbb{E}[Y]$ in the following case: $$ \mathbb{E}[Y]= \mathbb{E}\big[\min(X_1,\ ...
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181 views

Variance of Coin Flips Until H

If we flip a fair coin until we get heads, what is the variance of the number of flips to do this? My attempt is: $$E(flips):=Y=1\times P(H)+(1+Y)\times P(T)$$ $$\Rightarrow Y=\frac{1}{2}+\frac{1+Y}{...
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1answer
28 views

What does an expectation with respect to a policy mean in the reinforcement learning value function

I would like to know what the formal definition of the following expression is $$ V_\pi(s) = \mathbb{E}_{\pi}(G_{t+1} | S_t =s) $$ What does it mean to have the policy in the subscript? How would I ...
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2answers
45 views

Calculating inter-annotator agreement

Are there situations when it is allowed to omit calculating expected agreement but use only observed agreement as reliable measure? I have multi-label classification (in particular annotation of ...
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1answer
46 views

Estimating Expected Order Statistics

I have a fairly basic question that I'm looking for a reference for. First, a couple definitions. Let's say $X_1,\ldots,X_n$ are IID samples from a distribution $F$ over $[0,1]$. For any $k\in\{1,\...
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1answer
51 views

Find an expectation of normal random variable

How can I find an expectation of random variable $\xi \sim \mathcal{N}(a,1)$, where $a$ is random variable: $a \sim \mathcal{N}(0,\sigma^2),\quad \sigma = \text{constant}$ ?
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1answer
40 views

Elements of Statistical Learning training set [closed]

I am trying to read the Elements of Statistical Learning Tibshirani, Hastie and Friedman, however I have a problem with understanding the expected (squared) prediction error ($EPE$) formula that they ...
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22 views

Expected value of a geometric Brownian motion

Given a Brownian motion $B(t)$ and $X(t):= e^{B(t) -at} $ What does $\mathbb{E}(X(t))$ equal and why?
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2answers
81 views

Show that if $E\psi(x-\theta)= 0 $ then $P(X< \theta) \leq p \leq P(X \leq \theta)$

Define $$\psi(x)=\begin{cases} 1-p & x < 0 \\ 0 & x=0 \\ -p & x> 0 \end{cases}$$. I have to show that if $$E\psi(x-\theta)= 0 $$ then $$P(X< \theta) \leq p \leq P(X \leq \theta)$$...
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1answer
33 views

Expected value of residuals for LASSO model?

For simple OLS models the expected value of the residuals E(ϵ)=0 can be shown to be zero if an intercept is included in the regression equation. I am using a LASSO model and was wondering if the ...
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41 views

How to calculate Expectation of Multivariate Gaussian with x drawn from another Gaussian distribution?

Say, there is a $n$-dimension multivariate Gaussian, $g(x) = N(x:\mu, \Sigma)$ where $\mu$ is $n$-dim mean vector, and $\Sigma$ is $n \times n$-dim covariance matrix. I would like to calculate "...
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1answer
34 views

Is this method of finding the expected value of the square a random variable correct?

Suppose x is a discrete random variable with values 2,3,1 and probabilities 0.2,0.3, and 0.4 respectively. NOw say we have the function y=x2+3 and we want to find the expected value of this equation. ...
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1answer
46 views

What test to use to compare observed and expected frequencies when expected frequencies for each subject are independent from each other?

I conducted a study where I presented subjects with a treatment that they could either respond to with a match or non-match. I will use whales as an example. Whales can breach the water in several ...
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1answer
23 views

Population autocovariance goes to zero, assuming covariance stationary

In time series context, let $\gamma_j=E[(y_t-\mu)(y_{t-j}-\mu)]$ denote population autocovariance, where $\mu$ is population mean of $y_t$, assuming covariance-stationary. Then, $\gamma_j$ goes to $0$ ...
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55 views

Stuck on a term in $\operatorname{Var}\left[ \widehat{\beta}_0 \right]$ proof

So I was trying to prove that $\operatorname{Var}[\hat{\beta}_0] = \dfrac{\sigma^2n^{-1} \sum{(x_i)^2}}{\sum{(x_i-\bar{x}})^2}$ And I got stuck with the part $\dfrac{-2\bar{x}}{\sum{(x_i-\bar{x})^2}} ...
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24 views

Constant Terms in Linear Projection

In my time series textbook, it says, "Let $Y_i$ and $Y_j$ be two dependent variables in a time series process, e.g. $Y_{t+1}=\phi Y_{t}+\epsilon_{t+1}$, where $\phi$ is a constant coefficient. If a ...
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1answer
53 views

Expected value of squared least squares estimator

I am trying to prove $E(\hat{\beta} '\hat{\beta}) = \beta'\beta+\sigma^2 *\sum_{k=1}^K\lambda_k^{-1}$ where $\lambda_k$ denotes the eigenvalues of the matrix $(X'X)$ with dimensions $K\times K$. $\...
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43 views

Expectation of product

Let $\{X_i\}_{i\in I}$ be a finite collection of i.i.d random variables. I have found that $$E[\prod_i X_i]=\prod_i E[X_i]$$ But I haven't found a proof of this fact. What is the proof?
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Introducing a Bernouli random to ensure expecation remains the same after quantification

I'm going through a paper where they use a compressed floating point representation to save space when communicating the values over the network. Each floating point value $q$ is represented as a d-...
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2answers
28 views

Prove minimum argument of square error function is equal to expected value

Consider a probability density function $f(x)$ defined over the interval $[a,b]$ where $-\infty<a<b<\infty$. The square error function is defined as $J(y)=\int_a^b (x-y)^2f(x)dx$. The ...
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PCA: mean of marginal distribution of high-dimensional vector

Consider the following probabilistic model: $$p(x) = \mathcal{N}(0, I_d), \ x \in \mathbb{R}^d$$ $$p(y|x) = \mathcal{N}(Wx + \mu, \sigma^2I_D), \ y, \mu \in \mathbb{R}^D, W \in \mathbb{R}^{D\times d}...
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What is the expected value of the logarithm of Gamma distribution?

If the expected value of $\mathsf{Gamma}(\alpha, \beta)$ is $\frac{\alpha}{\beta}$, what is the expected value of $\log(\mathsf{Gamma}(\alpha, \beta))$? Can it be calculated analytically? The ...
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1answer
29 views

given $Z_i ~ N(0,1)$ and $Z^2 = Z_1^2 + Z_2^2$ what is $Cov(Z^2,Z_1)$?

Currently I am at the stage where: $Cov(Z^2,Z_1) = E(Z^2*Z_1) - E(Z^2)E(Z_1)$ $=E(Z^2*Z_1)$, because expectations of normal, or combination of normal variables is zero. After this I have no idea ...
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1answer
51 views

Advertisment decision making based on customer past behaviour

Problem description: Every 3 weeks a fashion company sends out an expensive booklet with descriptions of clothes to each customer on their electronic records. There exists a purchase history what each ...
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83 views

Proof Verification: $\tilde{\beta_1}$ is an unbiased estimator of $\beta_1$ obtained by assuming intercept is zero

Consider the standard simple regression model $y= \beta_o + \beta_1 x +u$ under the Gauss-Markov Assumptions SLR.1 through SLR.5. Let $\tilde{\beta_1}$ be the estimator for $\beta_1$ obtained by ...
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1answer
40 views

Expected Value of an AR(1) process

I saw the answer on this post and got confused about a couple things in its explanation. Mainly, I am unsure of How the poster immediately knows the process $X_t = c+\phi_1 Y_{t-1} + \epsilon_t$ is ...
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24 views

Relation between v(s) and q(s,a) in a Markov Decision Process?

I was solving questions related to backup diagrams from Reinforcement Learning: An Introduction by Barto and Sutton. Are these 4 equations mathematically correct ? Are there any shortcomings in terms ...
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91 views

Expected value of the residuals

How would one prove that the expected value of the residuals from OLS regression is zero? I will make two cases. In the first case I treat $X_i$ as random and in the second case I treat it is non-...
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1answer
27 views

Probability - expected value

The random variable $X$ takes on values -2, 0 and 2 with probabilities 1/4, 1/2 and 1/4 respectively. Find $\text{E}(X)$ and $\text{Var}(X)$. Till this part, it was easy enough. Then the question ...
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1answer
18 views

Simple Appplication of Law of Iterated Expectation

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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What is the expectation of ratio of two uniform distributions? [duplicate]

If x and y are distributed uniformly between 0 and 1. What is the expected value of their ratio ?
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53 views

Is $\bar X$ a random variable or a constant?

I am confused how $\bar X$ is used sometimes as a constant and othertimes as a random variable. My understanding is that $\bar X$ is a random variable because it changes every time our sample changes....
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1answer
36 views

Integral from the Adversarial Spheres paper (maximum of the difference between a constant and a normal random variable)

I'm trying to follow a proof in the Adversarial Spheres preprint on arXiv. The proof requires the computation of the integral in Appendix F, page 14: $$\mathbf{E}\left[\max\left(\sqrt{2}\left(\frac{\...
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18 views

Trying to Understand Power Series Expansion of Cumulant Generating Function

I'm having difficulty expanding the cumulant generating function, $K_X(t)$ as a power series. Specifically, I'm trying to go from the definition of the cumulant generating function, i.e. $$K_X(t) = \...
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46 views

For any three random variables $X,Y,Z$, prove or disprove that $\langle XY\rangle$, $\langle XZ\rangle$, $\langle YZ\rangle$ can't all be negative [duplicate]

For any three real random variables $X,Y,Z$, prove or disprove that $\langle XY\rangle$, $\langle XZ\rangle$, $\langle YZ\rangle$ can't all be negative. Here $\langle \cdot \rangle$ denotes an ...
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Can additional iterations of backward induction as described affect optimal policy?

Consider a game with the following properties: Single player Finite number of game states (after the player arrives at a terminal state, he or she can begin again from the start state; the player can ...
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53 views

Expectation of the function of random variable

How can I calculate the following expectation $\operatorname E\left[\int_0^1 t^{n+Y-1} \, dt \right]$ where $Y\sim \operatorname{Binomial}(S,\delta)$?
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32 views

Expectation of random sum of dependant variables

The expectation of random sum of independent identically distributed variables is given either by the law of total expectation or by Wald's identity. Are these generalised to tackle the random sum of ...
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1answer
24 views

Expectation Help

I am having trouble a question in my probability course and was looking for some insight into it. A digital clock sits next to your bed and in the mornings when you don't use an alarm you notice Y =...