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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Calculating the expected value of the log multinomial probit

I am stuck trying to solve the calculation of an expectation related to the multinomial probit likelihood. Say I have a random vector $\mathbf{F}$ whose components, $F_1$, $F_2$, $F_3$, are ...
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Law of unconscious statistician for conditional expectation

I have random variables $X$, $Y$ with joint distribution $f_{XY}(x,y)$ and conditional distribution $f_{X|Y}(x|y)$ and another random variable $Z=g(X)$ with $g$ being bijective is it true that $$E(Z|Y=...
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Decomposition of the second moment of a circular distribution

When we have a "usual" random variable $X$ on the real line, we can break down the second moment. $$ \mathbb{E}[X^2] = (\mathbb{E}[X])^2 + var(X) $$ Is there an equivalent for a circular ...
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Where does the expected value definition come from? [duplicate]

The definition of the expected value on the domain $[a,b]$ is given by $$E[X] := \int_a^b x f(x) \, \mathrm dx $$ I understand what the mean is, but I don't fully understand how this specific equation ...
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Comparing performance to random in multiple choice test

Background TLDR I was asked to grade some multiple choice tests. I noticed a difference between two groups (which differ by the part of our online materials they used). But I think it would be more ...
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What is the expectation of absolute difference

Given: E(|X-R|) =2.5 E(|L-X|)= 3 where X, R, L are assumed to be normally distributed and independent. Standard deviations of |X-R| and |L-X| are known. what is E(|L-R|) ?
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conditional expectation coefficients [closed]

We will consider projections of the form, Conditional Expectation: $$E[q_t|ε_t, ε_{t–1}, . . . ] = β_0ε_1 + β_1ε_{t–1} + β_2ε_{t–2} + . . . $$ $ε_t$ is white noise with mean $0$ and variance $σ^2_ε$ ...
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Expected value of X^3 for a normal distribution given it has limits?

What is the expected value of $X^3$ with in limits for a normal distribution? In other words, I am looking for solution of $E(X^3 \mid a\leq X \leq b)$.
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giving the sample covariance formula I want to show that they are equal [closed]

I want to understand how to move from the first formula of sample covariance in the image attached to the one below as indicated by the arrow
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1answer
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Laplace and Normal Distribution Cross Entropy

I need the following integral and struggle with calculating it or finding a citable source. $$\int_{-\infty}^{\infty}(x-\mu)^2\exp\!\left(-\frac{|x-\nu|}{\tau}\right)dx.$$ Background: I want to find ...
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Creating an expected dataset to compare with observed data to test for genetic interactions

I am trying to define a test to tell if two mutations combined give a greater phenotype than what you would expect by simply adding both. Here is an example (using R): ...
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Why is the expected value of variance different than the expected value of Maximum Likelihood variance?

The expected value of variance is: The expected value of the maximum likelihood of variance is: $E\left[\frac{1}{N}\sum_{n=1}^{N}(X-\mu_{ml})^{2})\right] = \frac{N-1}{N}\sigma ^{2} $ Why does ...
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Conditional expectation of product of sums of bernoulli random variables

let's say we have $X_1,..,X_n$ i.i.d. Bernoulli random variables. For $l<m<n$, we want to calculate: $$ E \left[ \sum_{i=1}^{l}X_i \sum_{j=1}^{m}X_j \mid \sum_{k=1}^{n}X_k \right] $$ Does this ...
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Modeling Expected Value Using Quantile Regression as an Ensemble

I'm trying to find a primer on a topic that I'm sure must have been studied, but I can't find anything on. Suppose we'd like to do regression in a supervised learning setting to learn the expected ...
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How is the relation of expectations and order statistics of X and Y, if Y is a mean-perserving spread of X

I am searching now for quite some time for an answer with a source to the following question. $X$ and $Y$ are two random variables. $Y$ is a mean preserving spread of $X$. Now, how is the relation of ...
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Expected rolls to roll every number on a dice an odd number of times

Our family has recently learned how to play a simple game called 'Oh Dear'. Each player has six playing cards (Ace,2,3,4,5,6) turned face-up, and take turns to roll the dice. Whatever number the dice ...
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1answer
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Question relating to joint PDFs

Here are my questions: Let $X$ ~ Unif$(0, 1)$, and $0<a<b<1$. Also, let \begin{cases} Y = 1 & \text{if $0<X<b$} \\ ...
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1answer
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Bayesian Statistics: Derivation of the expected value of the posterior predictive distribution

I am reading the Gelman, et. al., book Bayesian Data Analysis and they talk about the expected value of the posterior predictive distribution as follows. However, they did not really seem to derive ...
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What is meaning of taking the partial derivative of the integral of the gaussian with respect to variance?

In order to prove that: $E[(x- u)^2] = σ^2 $ Bishop takes the integral of the gaussian distribution (which equals 1): $∫ N(x|μ, σ) dx = 1$ Then he takes the partial derivative with respect to ($σ^2$) ...
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Expectation of Mixed Random Variable (Contradiction with Manual Solution)

$X \sim \mathcal{N}(1,\text{negligible variance})$ and $Y \sim \mathcal{N}(2,\text{negligible variance})$ \begin{equation*} Z= \begin{cases} X, & \ \text{w/pr}\quad p\\ Y, &...
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Show that $x_{t},y_{t}$ are jointly stationary, and interpretation of CAcovF, $\gamma_{XY}(h)$ not being symmetric for lags $h$

Consider two white noise processes $(w_{t})_{t}$~$WN(0,\sigma_{w}^{2})$ and $(u_{t})_{t}$~$WN(0,\sigma_{u}^{2})$ that are also independent of each other such that $y_{t}=w_{t}-\theta w_{t-1}+u_{t}$ ...
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What is the difference between using SDs away from the mean and SEs away from the expected value?

I've always learned about the bell curve concept as having units in SDs away from the mean. For example, if the mean is 5 and the SD is 2, then the curve is centered around 5, and, for example, a data ...
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Expectation of differences between arcs on a circle

Consider a circle with a circumference of $n$. On this circle, I define two arcs of length $k<n$, $A_1$ and $A_2$. The centres of the two arcs are uniformly distributed on the circle. Let $\Omega_{...
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Expectation (x/y) using jacobian

Let $X$ and $Y$ be two independent random variables with the density functions: $f(x) = 3 x^2$, for $0<x<1$, $0$ elsewhere $g(y) = 4y^3$, for $0 <y<1$, $0$ elsewhere Give $\mathbb E(x/...
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Identically distributed but not independent

Suppose X1 and X2 are identically distributed random variables, not necessarily independent, taking values in {1, 2}. If E(X1X2) = 7/3 and E(X1) = 3/2, obtain the joint distribution of (X1, X2).. So ...
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Extension of polya's urn

An urn contains r > 0 red balls and b > 0 black balls. A ball is drawn at random from the urn, its color noted and returned to the urn. Further, d > 0 additional balls of the same color are ...
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Is epsilon error a standard known error or custom created by this paper?

I'm reading this computer vision paper, research paper link, about creating a model to estimate the real age and perceived age of the person in the image (or at least that's what I think it's about). ...
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how to interpret the output of predict coxph with type=expected

The response in the link below clearly shows how the output of predict(data, type='risk') is calculated. I've been trying to find a similar explanation for predict(data, type='expected') with someone ...
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Expected Number of Trials to Terminate Random Sequence

I recently encountered this question and I'm having difficulty coming up with a rigorous explanation to back up the intuitive answer. Let $R$ be a random number generator such that $R(n)$ returns an ...
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Conditional expectation of a Gamma Distribution

$X$ a Gamma Distributed random variable. Calculate $E[X\mid X \in [a,b]]$, where $a>0$, $b>0$. Is there a closed form solution for this, and if so, how can I calculate it?
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Distribution where mean and mean of reciprocal are both 1 [duplicate]

Is there a non-trivial distribution for a positive r.v. $X$ such that $\mathbb{E}(X) = \mathbb{E}\left(\frac{1}{X}\right) = 1$? If possible, I'd like the distribution to support (have positive ...
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Confusion on the connection between causality and stationarity and possible implications

Let $(x_{t})_{t\in \mathbb Z}$ be a causal AR(p) process with operator $\phi$ such that $\phi(L)=\phi_{0}-\phi_{1}L-...-\phi_{p}L^{p}$ and $(\epsilon_{t})_{t \in \mathbb N_{0}}$ white noise sequence: ...
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Conditional expectation of a Weibull distributed random variable

Let $X$ be a Weibull Distributed random variable. I want to calculate $E[X\mid X \in [a,b]]$, where $a>0$, $b>0$. Is there a closed form solution for this, and if so, how can I calculate it?
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Notation about conditional expectation $E[Y|X]$

Given $X,Y$ real random variables, we know that $E[Y|X]$ is X measurable and that there is a Lebesgue measurable function $f : \mathbf{R} \rightarrow \mathbf{R}$ such that $E[Y|X]=f(X)$ almost ...
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Rigorous statement of expectations for the bias-variance trade-off

Consider a data generating process $$Y=f(X)+\varepsilon$$ where $\varepsilon$ is independent of $x$ with $\mathbb E(\varepsilon)=0$ and $\text{Var}(\varepsilon)=\sigma^2_\varepsilon$. According to ...
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1answer
36 views

Marginal distribution

A loss distribution has PDF - $f(x) = 1/x^2$, for $x > 1$ An insurer finds that the time in hours it takes to process a loss amount x has a uniform distribution on the interval $(\sqrt x, 2\sqrt x)$...
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Average of the outside of a truncated normal distribution

I have a sample that is normally distributed ~ $N(\mu,\sigma)$ and truncated between $a,b$ such that $a<b$. I saw a Wikipedia article that the average of the truncated part is $\mu + \frac{\phi(\...
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1answer
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Expected Mean using Total Law of Expectation

So I have an idea of conditioning on a new set variable say y that is 0 if the first toss is tails and 1 if heads although I am not sure how to structure this answer. I believe the law of total ...
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Intuition for why the (log) partition function matters?

I'm on a quest for the intuition behind the fact that theoretical introductions to approximate inference focus so much on the log partition function. Say we have a regular exponential family $$p(\...
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1answer
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Algebra of expectations in the MSE Decomposition

In the MSE decomposition formula why does the following hold? $ {\begin{aligned}{E} _{\theta }\left[2\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)\left(\operatorname {E}...
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Is this true:$ E[A|C]=E[A|B∩C]P(!B|C)+E[A|!B∩C]P(!B|C)$?

That is the conditional extension of $$E[A]=E[A|B]P(B)+E[A|!B]P(!B)$$ $$E[A|C]=E[A|B∩C]P(B|C)+E[A|!B∩C]P(!B|C)$$ allowed me to get the right answer. Thank you for the help.
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Bias-Variance decomposition: Expectations over what?

The Bias Variance decomposition is a decomposition of an expectation, but I fail to follow what's actually assumed random specifically in this decomposition. Take the specific regression example ...
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Distance in Square between Randomly Selected [duplicate]

I am trying to find the expected Euclidean distance between independent, randomly-selected variables in the unit square and I have some technical questions. For co text, I know if we were selecting ...
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1answer
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Expectation of any function of X, a nonnegative integer valued random variable [duplicate]

How to show that if X is a nonnegative integervalued random variable with distribution F,then $$E(X)=\displaystyle\int_0^\infty \overline{F}(X)dx$$ and $$E(X^n)=\displaystyle\int_0^\infty n*X^{n-1}\...
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Detailed proof of the Law of the unconscious statistician

Im trying to disentagle the proof of LOTUS but somehow got stuck in the last parts; this is the source on Quora which is pretty much the same as on Wikipedia. Here's my attempt to do it in more ...
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Conditional expectation and variance of a poisson distribution, where $𝑋_𝑛$ is conditioned on $𝑋_{𝑛−1}$ [closed]

Suppose that $𝑋_0 \sim \text{Poi}(𝜇)$ (where $𝜇 > 0$ is fixed) and that for $𝑛 = 1, 2,…$ the distribution of $𝑋_𝑛$ conditional on $𝑋_{𝑛−1} = 𝑥$ is Poisson with parameter $𝑥$. Determine ...
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Where to learn about expectation

Now, this might sound a little weird, but let me explain. When one has a random variable (like an estimator), one often cares about two things. The first is concentration -- now, I have seen many ...
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1answer
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Expectation of division of random variables [closed]

Is this true? E(Xn/Yn) goes to E(Xn)/E(Yn) in probability even if Xn and Yn are not independent?
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Expected Predicted Error (EPE) with L1 loss

In Element of Statistical learning it is saying on page 20, equation 2.18. That using the L1 norm instead of the usual L2 norm leads to an $f(X)$ optimising the EPE being the median instead of the ...

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