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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Continous version of an (almost) negative binomial distribution

Let a random variable $X$ be the number of independant Bernoulli trials needed to reach $s$ successes and $f$ failures when the probability of success is $p$. We therefore stop trials when we have $s$ ...
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Complete expectation of life in Demography

This is the question that I am trying to solve , I have done it halfway , but I am unable to comment on the nature of linear relationship between x and complete expectation of life
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If $E(X|Y) = X$, what does this imply about the relationship between $X$ and $Y$?

Let $Z, K, V$ denote random variables, where $Z$ is binary, $K$ is categorical from 1-10, and $V$ is continuous. Let $X = P(Z = 1|V = v)$ and $Y_k = P(Z = 1|V = v, K = k)$. Now define $Y = (Y_1, \...
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Bounding sum of quartic deviations from sample mean

[Cross-posted here with no answers for a few days] I came - to the very best of my knowledge from reading the source - across the following statement in The Jackknife and Bootstrap, Shao and Tu, p. 87:...
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Is the expectation of a sample of independently and identically distributed random variables the expectation of just one of them> [closed]

What the title says. Since each 𝜉𝑖 is identically distributed, would it be correct to assume that the expectation and variance of one of them is the same as that of every other one? Would the ...
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Moments of Linearly Transformed Laplace Distribution and Assumed Density Filtering

I have been occupied with a question that I assume is not as difficult as I find it to be. The question I want to solve boils down to finding the moments of a linearly transformed Laplace distribution....
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Probability that the expected value (mean) is within one standard deviation of an exponential distribution [closed]

fx = 1/5e^-x/5 x>0 Calculate the probability if the mean is within one standard deviation. What I Know: mean = 5 (because it is theta or the inverse of lambda from the fx) In an exponential ...
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Expectation with respect to a product distribution

Let $\theta \in \Theta$ be a $d-$dimensional random variable. Let $q$ be a distribution on $\Theta$ of the form $$q(\theta) = \prod_{i=1}^d q_i (\theta_i).$$ In other words $q$ is a product of ...
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Expected value and variance of Autoregressive model

I stumbled against this problem and found it hard and help would be much appreciated We have the following equation $Y_{t}-Y_{t-1}= 1/3(Y_{t-1}-Y_{t-2})+a+u_{t}$ Where $a$ is a parameter and $u_{t}$ ...
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Conditional Expectation : How is E[E[xy|x]]=E[xE[y|x]]?

I read that "The law of total expectations states E[xy]=E[E[xy|x]]. By linearity of conditional expectations, E[E[xy|x]]=E[xE[y|x]]" but I am not able to understand the part "E[E[xy|x]]=...
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Find the expected number of steps [duplicate]

Consider the following problem: Two people $M$ and $T$ walk over a straight line. The steps they take depend on flipping a coin: They move to the left if the result is head and they move to the right ...
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Expected value of the product of random sums [closed]

I have the following set of independent poisson distributed variables (for all $i$): $$N_{i,k}\sim \mathcal{P}(n_k)$$ $$N_{i,p}\sim \mathcal{P}(n_p)$$ and $Z_p$ a random variable with expected value $...
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Calculate mean of $X$ when $X$ and $Y$ are jointly normal and $Y$ is truncated above [duplicate]

Suppose I have two random variable $X$ and $Y$ and they are distributed joint normally and $Y$ is truncated above by constant $c$ $$\begin{pmatrix} X \\ Y \end{pmatrix} = TN\left(\underbrace{\begin{...
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Computing difficult expectation

I am doing some derivations for a project and at some point the following integral shows up $$ \int x^{c} \log(1-F(x)) N f(x) [1-F(x)]^{N-1} dx $$ i.e the expectation of the product of natural ...
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Expected winning prize of 95% of total prize amount based on Probability of not winning and Probability of winning for each individual

For this probability model, As I am new to probability, I am looking for an approach of the underlying probability concept to work on the calculation in arriving at the expected winning prize of the ...
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What is the expected value and variance of the arithmetic mean

How can I calculate the expected value and variance of the arithmetic mean? I know that is the arithmetic mean, but how do I use this to figure out the expected value?
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How to interpret mean of this estimated AR(1) process

I estimated an AR(1) process, my data looks like this: Making usual unit root test, they suggest that an estimated AR(1) from this data is stationary. Estimating the AR(1) over this data, these are ...
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AR Unconditional vs Conditional Moments

I was hoping someone might be able to explain the intuition behind conditional and unconditional moments and specifically when I ought to us which. In the sense that: I first started with considering ...
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Expected value of $XY^2$ or $Cov(X,Y^2)$

I wanted to find the solution to $E[XY^2]$ where $X$ and $Y$ are random variables with given mean and variance. How can I find $Cov(X,Y^2)$?
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Expectation of Maximum and Minimum of Partial Sums of Normal Random Variables

Peggy Strait, 1974, Pacific Journal of Mathematics ON THE MAXIMUM AND MINIMUM OF PARTIAL SUMS OF RANDOM VARIABLES Gives a nice result (4.3) and (4.4) in terms of "standard normal random variables&...
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Interpretation of Huygens Expectation

Christiaan Huygens wrote in "Libelus de Ratiociniis in Ludo Alae" (can be found here: https://math.dartmouth.edu/~doyle/docs/huygens/huygens.pdf page 2, Postulat) : "That any one Chance ...
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Finding mean of perturbed Gaussian using perturbation theory

I am trying to approximate the mean of the following distribution using perturbation theory: $$P(x) \propto \exp\left( \frac{1}{2\sigma^2}x^Tx + \alpha\frac{1}{2}(f(x)-y_0)^2 \right) $$ where $y_0$ ...
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Concrete bound of expected value of a difference of I.I.D. Uniform Random Variables

In the following, $X,X_1,X_2,\dots X_n$ are I.I.D. uniform random variables in $[0,1]^d$ in $\mathbb{R}^d$. The problem I am attempting to solve is Exercise 2.4 from Gyorfi's "A distribution free ...
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What are E(max(X1, X2)) and Var(max(X1, X2)) when the Xs are normal random variables? [duplicate]

Let X = (X1, X2) be normally distributed random variables with mean m = (m1, m2) and covariance matrix S. Y = max(X1, X2) = X1 + max(0, X2 - X1) = X1 + D (X2 - X1), where D = 1 if X2 > X1 and 0 ...
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Fisher Information Matrix of Matrix Variate F Distribution

Let $\mathbf{X}$ follow a matrix variate F distribution with pdf $$ \begin{align} f\left(\mathbf{X} | \mathbf{\Sigma}, n, \nu\right) = \frac{\Gamma_p(\frac{n + \nu }{2})}{\Gamma_p(\frac{n}{2})\...
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Calculation of Variance from a 2 order Taylor expansion - Expecting a better estimation than with 1st order Taylor expansion

I tried to compute the variance of a squared ratio of 2 Gaussians random variables (not the same means and standard deviations between both). I generate the samples by Monte-Carlo method. I expect ...
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Why is this integral equal to $1$? (VBIL)

Let $p(y \mid \theta)$ be a likelihood and $\hat{p}_N(y \mid \theta)$ be an unbiased estimator of it. In VBIL they define $z = \log \hat{p}_N(y \mid \theta) - \log p(y\mid \theta)$ and call its ...
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Expected Value Contradiction

I am stuck with a simple expected value problem. Here it is: Assume I have 1 USD in savings now. I have to decide whether I keep my savings in USD or convert them into Euros. At the moment 1 USD = 1 ...
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Expectation of the log shifted-gamma

If $X \sim \Gamma(\alpha, \beta)$ and $c$ some constant then $Y=X+c$ follows a shifted Gamma distribution with pdf $$f_Y(y)=\frac{b^a}{\Gamma(a)}(y-c)^{a-1}e^{-(y-c)b}$$ $y\in[c, +\infty)$. What is ...
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Differentiating expected prediction error (EPE)

From Hastie-Tibshirani-Friedman p.18-19 $$ EPE(f)=E(Y-f(x))^{2} = \int[y-f(x)]^{2}Pr(dx,dy) $$ If $f(x)\approx x^{T}\beta$ shows that by plugging in $f(x)$ in $EPE$ and differentiating w.r.t. $\beta$ ...
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Expected first passage time for random walk

A random walk on $\{0,1,2.....n\} $ with $p_{0,0} = p_{n,n} = 1$ and $p_{i,i+1} = p = 1-p_{i,i-1} = 1-q$ for $1 \leq i \leq n-1$ .Let $X_0 = i $ and $T$ be the first passage time to either 0 or $n$ ...
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Express expectation value of a joint distribution over a discrete and continuous random variable

Let $Y$ be a discrete random variable and let $X$ be an (absolutely) continuous random variable and $f(X, Y)$ a function of these two random variables. Let $P(X, Y)$ be the joint probability measure. ...
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Expected Value of a Fundraiser Raffle Ticket with Three Prizes

The raffle ticket above represents a local fundraiser. The first prize is a smart TV valued at 1200 dollars, the second prize is a cooler valued at 200 dollars, and the third prize is a 100 dollar ...
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Finding expected value from expectation of squared distance

This problem is actually a part of a much larger biology problem that I am working on. However, I will leave out the unrelated parts. Consider a sequence of points $\{(x_j, y_j)\}$ where neighboring ...
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Equivalence of Inverse Probability of Treatment Weights and Standardization: Hernan and Robins proof

Hernan and Robins provide a proof for the equivalency of inverse probability weights and standardization for estimating the potential outcome mean that I am struggling to follow (technical point 2.3, ...
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What distributions have an undefined mean but are not symmetric?

What distributions have an undefined mean but are not symmetric? I'm looking for a probability distribution function (and CDF) for which the mean is undefined, but not symmetric like Cauchy, but a ...
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Implications on the relation between signs of random variables

Consider a binary random variable $Z$ and a random variable $Y$. Suppose that the following relations hold $$ Z=1 \Rightarrow Y\in \mathbb{R}^{+}_0\\ Z=0 \Rightarrow Y\in \mathbb{R}^{-}_0 $$ In words, ...
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Calculating $\mathbb{E}^2(\sigma_t^2)$ where $\sigma$ is a GARCH(1,1) process

Given that $\alpha=0,113079$, $\beta = 0,873884$, $\omega = 0,0000081$ (and that $\text{kurtosis} = 235$), I need to calculate a call price using GARCH volatility: https://www.researchgate.net/...
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Simple mean or kernel estimator?

Let $X$ be a continuous random variable with support $\mathcal{X}$ and density $f(x)$. Suppose I'm interested in constructing a consistent estimator of $E(X)$ using $n$ i.i.d. observations $(X_1,..., ...
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What is the expectation of $e^X$, where $X$ is a random variable with a geometric distribution?

if $X$ is a random variable how can I calculate $$ E(e^X) $$ I have no idea on how to do that.
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Expectation of sequence of Random Variables

$X_1, X_2, .....X_n$ is a colletion of Random Variables. They are said to be $\textit{multiplicative system}$ if, for any $1 \leq k \leq n$ and for any set of $k$ indices $1 \leq i_1 < i_2 ....i_k \...
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rank of an expected value of a matrix

x is (a * 1) vector y is (b * 1) vector x and y are independent then what is rank(E[xy']) I know that xy' should be (a*b) matrix and since they are independent. , however I am not sure about the rank
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The expected value of log Gamma function

Suppose $X$ is exponentially distributed with the rate parameter $\lambda$. If we have the expected value of $\log X$ as \begin{equation} \langle \log X\rangle=-\gamma-\log\lambda \end{equation} where ...
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Expected value of the sum of Bernoulli r.v. with p = cdf of Normal r.v

Let $y_1, y_2,\ldots, y_n$ be an independent samples from an unknown distribution and $q_p$ be its $p$th percentile. I constructed $B_i$'s such that $B_i \sim \text{Bernoulli}(F(y_i))$ where $F$ is ...
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The average of a random varible with pdf in the form of an parametric inegral

The pdf of a random variable $T$ in the interval $(0,1)$ in a certain problem I was trying to solve is given by : $$ g(t)= c\int_{0}^{1-t} t^{m-1}\left[(u+t)^{m}-u^{m}\right]^{n-2}(u+t)^{m-1} d u $$ ...
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Information Matrix for Conditional Likelihood

I am studying the MLE theory on my own and I am confused by the difference between the fisher information matrix for the full sample and for one observation, when it comes to conditional likelihood. ...
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Self Study: Trivariate Normal Expectation with Inequality Condition

I'm reading a paper and found an interesting expectation. I know the result the author found but I can't figure out the intermediary steps because the author provided none. My attempt is getting ...
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Order Statistics: How to calculate expected value of a function involving first and second order statistics

I am currently stuck with a challenging problem. I have n values drawn i.i.d. from a distribution F(x). Let $v_1$ be the nth order statistic (highest value) and let $v_2$ be the n-1 order statistic (...
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Intuition of Random Walk having a constant mean

I am very new to time series analysis. A random walk is defined as $Y_t=\phi Y_{t-1}+\varepsilon_t$, where $\phi=1$ and $\varepsilon_t$ is white noise. It is said that process is non-stationary for ...
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Calculating expected value for a gacha game

I'm trying to do some sanity check on a gacha game I'm playing as I suspect their calculations are off. I've already done a monte-carlo simulation to verify my hypothesis, but I'm not sure how to do ...

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