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Questions tagged [expected-value]

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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How to compute the expected number of events in the following conditional renewal process?

I have a stochastic point process with event times $\{x_1, x_2, ...\} $ and I want to compute the expected number of events $n(T)$ over the interval $[0,T]$. The point process is generated as follows: ...
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How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
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Expectation of a product of random variables problem

Let $U$ be a random variable with uniform distribution over $[0,1]$ and a bivariate random vector $(Z,T)$ defined by $$(Z,T)=(0,0)1_{\{U\geq 2\alpha\}}+(Q_X(U),Q_Y(U))1_{\{U< 2\alpha\}}$$ where $...
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How to model uncertainty in a binary choice problem

Background Suppose a respondent (id) is asked to make a binary (discrete) choice,i.e., either select option 1 or option 2 (choice) in five tasks (t=1,2,3,4,5) (a panel dataset with five observations ...
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How is this equation about expected value true?

$X$ is a stochastic variable. We have $E(X) = \mu$ and $Var(X) = \sigma^2$ Then $E(X^2) = \mu^2 + \sigma^2$ We have data $X_1, X_2, X_3 , \ldots, X_n$ from that distribution. Then we have: $$\sum_{...
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State value function $v(s)$ as an expectation

Suppose we have a Markov Decision Process with environment states $s \in S$, agent actions $a \in A$, and rewards $r \in \mathbb{R}$.So if an agent takes an action $a_t$ in state $s_t$, he will end up ...
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law of iterated expectations doubt

Is it correct to do so: $E(X^{-1} Y) = E_x(E(X^{-1} Y \vert X) = E_x(X^{-1} E(Y \vert X)) $ Are we allowed to use LIE in case of non linear functions like $X^{-1} $
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Deriving Value Function of a Markov Reward Process

I am looking through the coursework for a Reinforcement Learning course (I am not enrolled in it, this is for my own study). In the lecture notes, on page 6, equation 11, they provide the following ...
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expected value (Sheldon Ross 9th edition)

A person buy 10 lottery tickets with each having a winning probability p. If he wins a prize in atleast one of the tickets,he gets addicted and will keep on buying tickets till he wins again. Find the ...
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unbiased estimator using Walsh Averages?

In class, the professor said the median of the Walsh averages, $\mu _w$, can be used as an estimator. Further $$\dfrac{\textrm{variance of}\;\mu _w}{\textrm{variance of} \;\bar {Y}} =\dfrac{1}{ARE} ...
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Expectation of a reciprocal of unbiased estimation [duplicate]

Consider the two "true" value of directions $\bf{n}$, $\bf{s}$ and their "unbiased" estimate ${\bf{n}}_0$ and ${\bf{s}}_0$. Let ${\bf{w}}={\bf{n}}^T{\bf{s}}/{\bf{n}}_0^T{\bf{s}}_0$, then the ...
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Showing the expectation of a lognormal AR(1) process

Suppose I have a lognormal AR(1) process: $$\log(y_{t+1}) = (1-\theta)c + \theta \log (y_t) + \varepsilon_{t+1},$$ $$\varepsilon \sim N(0,\sigma^2)$$ To show $\operatorname{E}(y_{t+1})$, is it ...
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41 views

What does E_n mean

I came across the following symbol in this paper: I am a little bit confused by the symbol . Does it simply mean population average?
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32 views

Expectation of absolute difference between a random variable with its median [closed]

Given a random variable $x \sim p(x)$. Denote $m$ to be the median of x. What is the expected value of $|x-m|$, i.e we would like to compute the following integral: $$\int |x-m| p(x)dx $$
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Derivative of expectation where the variable appears in the integration limit and in the integrand?

I want to calculate the derivative of $$\varphi(\mu) = \int_{-\infty}^{\mu} r(x-\mu) f(x)dx,$$ wrt to $\mu$, where $r$ is a function and $f$ is a density function. How can I account for the presence ...
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on some bounds on expectation of probability measures on discrete set

I have a situation where I have one to one correspondence between $\mathbb N$ and a subset of probability measures on $\{1,2,3,\dots, m\}$ i.e $n\mapsto p_n$, suppose for all $N, M\in \mathbb N$ $\|...
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Distribution of integral values of monotonic functions [closed]

Consider the family of all integrable monotonically increasing functions $f:[0,1] \rightarrow [0,1]$ (which can be seen as cumulative distribution functions): Of a randomly picked function you know ...
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1answer
47 views

Absolute moment and integration by parts

Let $X$ be a real continuous random variable with distribution $F$ with finite moments. I want to calculate $$E[\vert X \vert] = \int_{-\infty}^{\infty} \vert x\vert dF(x)= -\int_{-\infty}^{0} x dF(x)...
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An elementary proof of the equivalence of measure theoretic and density expected values

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $X\colon \Omega\to\mathbb{R}$ be real-valued and measurable. Suppose there exists $f\colon \mathbb{R}\to [0,\infty]$ such that $P(X\in A)=\...
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What's $E[X - E[X | Y] |Y]$

I'm trying to figure out what $E[X - E[X | Y] |Y]$ equals to. Assuming that $X$ and $Y$ are continuous random variables, I thought that $E[X - E[X | Y] |Y] = E[X |Y] - E[E[X|Y]|Y] = E[X|Y] - YE[X|Y]$ ...
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Notation: What does the tilde below of the expectation mean? [duplicate]

I am reading about variational auto encoders, and there is the below loss function: $$l_i(\Theta,\phi) = - {\mathbb{E}}_{z\sim q} \left[\log p_\phi(x_i|z)\right] + KL(q_{\phi}(z_i|x)||p(z))$$ What ...
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Is the cumulative distribution function evaluated on the expected value $\frac{1}{2}$

Let $X$ be a random variable with density function $f(x)$, so $$ \int_{-\infty}^{+\infty}f(x)dx=1 $$ and $$ E(X)=\int_{-\infty}^{+\infty}{xf(x)dx}. $$ Then, my question is if $$ \int_{-\infty}^{E(X)}{...
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Conditional Expectation of Poisson

Suppose $X_1$,$X_2$,$X_3$,.....,$X_n$ are i.i.d. random variables with a common pmf poisson(λ) (t = a value) How would you calculate the below without using intuition (I would appreciate if you ...
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1answer
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Expected number of overlapping substrings

I want to know what the expected number of overlapping substrings there are when sampling with replacement from a large string. Suppose there is a string of length $N$, and we want to sample $m$ ...
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27 views

Why can we expand terms with random variables in the variance formula?

$\newcommand{\E}{\mathrm{E}}$$\newcommand{\Var}{\mathrm{Var}}$In the proof for showing the alternative formula for variance, i.e. $$\E[(X - \mu)^2] = E[X^2] - E[X]^2$$ I typically see the following ...
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What does the integral of a function times a function of a random variable represent, conceptually?

I am trying to understand conceptually what does the following give me or tell me: $$\int f(x) \cdot g(x) \, dx$$ where $f(x)$ is any continuous function of $x$ and $g(x)$ is the probability ...
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Multiply CDF by constant, what is the expected value of this “new” CDF? [closed]

Specifically, I want to multiply $F_X(x)$ by $E(X)$, so I have $$ ??? = E(X)\cdot F_X(x) = \int^b_a xf_X(x)dx\cdot \int^b_a f_X(x)dx \overset{?}{=}\int^b_ax\Big(f_X(x)\Big)^2dx $$ Is there a way to ...
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1answer
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Expectation of a sequence of gaussian variables

Suppose we are given a sequence of iid Gaussian random variables $\{x_k \}$ with zero mean and unit variance. We create a new random variable $$X_n = e^{\sum_{n\ge k}x_k}.$$ How does one go about ...
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Can the first difference of a time series follow a random walk?

I have a time series in levels, say $X_t$ and a variance-ratio test suggests that it does not follow a random walk. Now I differenced the time series once and the variance-ratio test suggests that the ...
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Expectation of Sufficient Statistic for Beta Distribution

I am looking at question 1b of the following notes: http://www.gatsby.ucl.ac.uk/teaching/courses/ml1/asst1.pdf In 1a, I have shown that the Beta distribution has a density that can be written in the ...
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What is the expected value of half a standard normal distribution?

You have a normal distribution with mean of 0 and variance of 1. Keeping the same probabilities and focusing only on half of the distribution (other half has it's original probabilities but x values ...
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Understanding the GAN Loss function from the original paper

I've been reading the paper Generative Adversarial Nets by Ian J. Goodfellow et al., to have a more deeper understanding about the concepts from the author's perspective (I do understand the basics of ...
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1answer
40 views

On a mistake computing the Kullback Liebler Information Criterion

THE FRAMEWORK: Let $X_1$ be an observation from a normal random variable with mean zero and variance $\sigma^2$ and lets call the PDF $f(x)$. I want to minimize the Kullback Liebler Information ...
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1answer
32 views

Estimating the population median from a kernel density estimator

I have a 1-d kernel density estimate in the form of two vectors: x_grid is a vector of x-values at which the density function was sampled ...
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1answer
40 views

Showing estimator is biased without assuming $X^TX$ is invertible?

I would like to show that the ridge regression estimator: $$\beta^R = (X^TX+\lambda I)^{-1}X^T Y$$ is biased, where $Y \sim N(X\beta, \sigma^2 I)$. If we assume that $X^TX$ is invertible, this can ...
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1answer
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Conditional expectation in Basic Linear Unobserved Effects Panel Data Model

I want to see that $ E[u_t|X_t,c] \Rightarrow E[X_t' u_t]$ in a Panel Data Models
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1answer
55 views

Why does differentiating the normalization term of a Binomial Distribution yield the expected value? [closed]

In Bishop's book Pattern Recognition and Machine Learning, problem 2.4 aims to derive the expected value of the binomial distribution $m \sim Bin(N, \mu) = {N \choose m} \mu^m (1 - \mu)^{N-m}$ by ...
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1answer
35 views

How can I prove the following relation between the probabiloity of X and its expectation using Cauchy-Schwarz inequality?

For a random variable $ X \geq 0 $ and $E[X^2] < \infty $, I'm asked to prove the following: $ P(X> 0) \geq \frac{(E[X])^2}{E[X^2]}$ It makes intuitive sense to me that it must be the case, ...
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where can find good and in depth explanation for expectation, manipulation of expectation, sampling instead of expectation?

I look for a book or online source, for better understanding the expectation, expectation inside the expectation or sampling for calculation of expectation. for example, in Richard S. Sutton's ...
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Product of a linear function of IDD variables [duplicate]

if $r_i$ are IID variables with $E[r_i]=μ$ and $f$ is a constant, is the following correct? $E\left[\prod_{i=1}^n(1+f\space r_i)\right] = \prod_{i=1}^nE\left[1+f\space r_i\right] = \prod_{i=1}^n(1+...
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How do we prove this identity related to expectation and variance?

Prove that if $\textbf{a}$ is a vector of constants with the same dimension as the random vector $\textbf{X}$, then \begin{align*} \textbf{E}[(\textbf{X} - \textbf{a})(\textbf{X} - \textbf{a})^{\prime}...
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IQN: Confusion about distortion risk measure

In the paper "Implicit Quantile Networks for Distributional Reinforcement Learning", they define $$ \begin{align} Z_\tau&:=F_Z^{-1}(\tau)\tag 1\\ Q&:=\mathbb E_{\tau\sim U([0,1])}[Z_\tau]\tag ...
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1answer
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Distribution of N objects into C bins that are then sorted?

Let's say we have $C$ bins and $N$ indistinguishable objects. For each object we choose one bin at random where each bin is equally likely (with probability $1/C$). Let $B_k$ be the number of objects ...
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Lottery with known EV and standard deviation. Probablity of positive ROI after x trials?

Lottery with the following payout table: **Payout** **Frequency** 0x 0.59992 1x 0.34992 5x 0.04982 1000x 0.00035 ...
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What kind of algorithms are appropriate for this sort of medium-dimensional integration problem?

I'm trying to model a situation in which an agent must select one of several choices (not more than ten). Each choice is associated with a vector, known to the agent, representing its effectiveness in ...
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1answer
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Expected value notation in GAN loss

I am reading Goodfellow's original paper on GANs. What I struggle to understand is his notation of the subscript in expected values. $\mathbb{E}_{\boldsymbol{x} \sim p_{data}(\boldsymbol{x})}$ .... ...
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1answer
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Find the expected value of choosing 3 balls among 8 (each balls has different value) [closed]

Let's say you have 8 balls in front of you, each of them contains a bank note (for example 1\$, 5\$, 10\$ etc...). And let's say you can choose randomly 3 balls among the 8 (you don't know what's ...
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2answers
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notation in standard form of covariance formula

I'm trying to build my basic intuition on covariance, but am confused about the notation I typically see: cov(X,Y) = E[(X - E[X])(Y - E[Y])] Why is there no summation sign in this formula, and why ...
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Importance sampling with partly deterministic samples

I am trying to estimate an expectation with a certain set of unlikely, but important (for the value of the expectation) events. In particular, let’s say I have a (normal) distribution p(x). Now, I ...
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1answer
31 views

Modified coupon collector: killing traitors problem

You have $n$ people, $m$ of whom are traitors. You keep killing them randomly until all the traitors are gone (which you presumably have some way to verify). On average, how many people $X$ will you ...