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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Expectation values [closed]

What is the expectation value of the expectation value. How to calculate: <∆j> where, ∆j=spread of data= j-<j.> ;for any variable j
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Is this a valid operation for an estimator? $$ E(\hat P ̂^2 ) = E((\frac{X}{n}) ̂^2 ) = \frac{1}{n^2} E(X^2) (?)$$ [closed]

Lets say the following estimator is defined: $$ \hat P = \frac{X}{n}$$ Where $$ X \sim B(n,p)$$ $$ E(\hat P ̂^2 )$$ Can you just substitute it like this: $$ E(\hat P ̂^2 ) = E((\frac{X}{n}) ̂^2 ) = \...
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How to show that simple random sample sensitivity is unbiased for population sensitivity

In diagnostic testing, sensitivity $S$ is the probability that the test gives a positive result given that you have the condition being tested. From a simple random sample of people who take the test, ...
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$E(SN)$ for aggregate claim amount $S$, $S=X_{1}+...+X_{n}, X_{i}$ are iid [duplicate]

Consider the following model for aggregate claim amounts $S$: $S=X_{1}+X_{2}+...+X_{N}$ where the $X_{i}$ are independent, identically distributed random variables representing individual claim ...
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Definition of Unbiasedness in Introductory Stat

If we look at Wooldridge's Introductory Econometrics, he has the following definition for unbiasedness of an estimator. Let $W$ be an estimator of $\theta$. $W$ is an unbiased estimator of $\theta$ if ...
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In a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$?

Sorry if obvious but in a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$? I don't really get what the random variable $x_t|x_t, x_{t-1},...$ represents? What I find ...
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Calculating expected value

Came across an interesting problem. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it to the ...
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Notation of expectation with conditional in subscript

Inside the book "The Elements of statistical learning", I stumbled upon the following notation (Ex. 2.7) $$E_{\mathcal{Y|X}}(f(x_0) - \hat{f}(x_0))^2$$ where $\mathcal{X, Y}$ are two random ...
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Expected value of a random variable with truncation

Let $f:[0,\infty)\to \mathbb R_+$ denote the PDF of a random variable $X$ and $c>0$ a constant. I want to evaluate the following integral: $$I(c)=\int_0^\infty{\min(x,c)f(x)dx}.$$ This can be ...
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What is the conditional expectation of two random Poisson variables?

Say I have two random variables $Z^{p}_{i} \mid Y^{p}_{i} \sim \text{Poisson} \left(t^{p}_{i}Y^{p}_{i}\right)$ and $Z^{q}_{i} \mid Y^{q}_{i} \sim \text{Poisson} \left(t^{q}_{i}Y^{q}_{i}\right)$ for ...
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Difference between geometric distribution expectation and 1 - failure with Binomial

I'm trying to understand a simple problem: How many times you'd need to roll two dice to get two ones in a single roll. One way I see this is as a problem the geometric distribution describes. You ...
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calculating expected value of false discovery rate - Benjamini-hochberg

I recently got a question that cannot seem to solve, as a beginner in the statistical field. assume we have 20 tests. 5 with the p-value of 1/330 5 with the p-value of 1/150 10 with the p-value of 1/...
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What does the subscript in an Expectation operator mean in the context of the bias-variance tradeoff?

I've been looking at the derivation of the bias-variance decomposition for a few months now and though I understand its implications, I am still confused by the notation used in the literature. ...
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Determining if a die is fair or not by rolling six times and observing only the sum

Suppose you have two bags and each has 6 dice. In one bag all the die are fair. In the other each die has a bias - die “1” has a probability of returning 1 of $\frac{1}{6} +\epsilon$ and $\frac{1}{6} -...
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How to compute (or upper bound) the expected value of power of the sum of product of Gaussian iid random variables?

Let $g_{1t}, g_{2t}, \ldots, g_{nt}$ are i.i.d. random samples from standard normal distribution $N(0,1)$, where $n$ is an integer and $t\in [T]$. For any $k>2$, how to compute the exact value or ...
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Conditional expectation calculation with a binary variable

Suppose $X=\{0, 1\}$ is a binary random variable, $X$ and $Y$ are independent, $Z$ is another random variable. I get \begin{equation} E\left(\frac{X}{E(X)}Z|Y, X\right)=\frac{X}{E(X)}E(Z|Y, X). \qquad\...
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Expectation of the ratio of sum (XY) and sum(X)

I want to know (mathematically) how the following expression changes as $M$ increases but still have no clue after thinking about it for a while. Any suggestions or comments will be much appreciated. $...
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what is the expectation value of a norm of a random variable from the standard normal distribution?

Let's suppose $\zeta_k \in \mathbb{R}^p$ is a random variable from a normal distribution with mean zero and 1 standard deviation. why $E[|| \zeta_k ||^2]=p$?
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Expectation of the product of mean independent random variables

I have random variables $a$ and $b$, which are such that $\mathbb{E}[a|b]=0$. I am trying to compute $E[ab]$. Is the following correct? $$\mathbb{E}[ab]=\mathbb{E}[\mathbb{E}[ab|b]]=\mathbb{E}[b\...
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Bayesian Quadrature to find expectation of unkown function w.r.t. known pdf

I am interested in estimating the integral $\int f(x) P(x) dx$, where $f(x)$ is an expensive function and $P(x)$ is has an analytic form. I would like to evaluate this with as few evaluations of $f(x)$...
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Suppose I have 100 integers and I sample 10 without repetition. What is the expected rank of the lowest out of 10 samples?

Suppose I have 100 integers and I sample 10 without replacement. What is the expected rank of the lowest out of 10 samples? i.e. my lowest integer in the 10-sample is kth smallest out of 100. what is ...
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Loss Size Index Function of A Gamma Random Variable

I'm trying to prove that the loss size index function of a Random Variable Y, which is distributed as a Gamma Random Variable ($Y \sim Γ(γ,c)$) has the following expression: $$ I(y) = \frac{\textit{G}(...
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Interpretation problems of linear model with no predictors

Let $Y=X\beta$, where $X=(\begin{matrix}1 & 1 & 1\end{matrix})^T$, $Y=(\begin{matrix}6 & 5 & 4\end{matrix})^T$ and $\beta=(\begin{matrix}\beta_0 \end{matrix})$. Now $X\beta=(\begin{...
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E(log(x)) to E(x) [duplicate]

Sorry if this is a straightforward question, but I have tried digging into econometrics book and cannot find anything about it. I worked on a model with ...
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Is it true that $\mathbb{E}_{X}[f(X)] = \mathbb{E}_{X, Y}[f(X)]$?

Suppose $p(x, y)$ has $x$-marginal $p(x)$ and that $f(x)$ is a well-behaved function. Is it always true that the expectation with respect to the joint distribution is equal to the expectation with ...
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ADAM bias correction derivation

Where $\beta_2\in [0,1)$ and $g_i$ is the gradient at step i. We approximate E[gi^2] with E[gt^2] by a correction term $\zeta$. I think this is because it is assumed that the gradients are bounded. Is ...
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Expectation of a random variable with positive density at infinity?

Let $X$ be a continuous random variable with pdf $f(x)$. Assume that $f(x) \geq 0$, for all $x > 0$, so $X$ is a positive random variable, $f(\infty) > 0$, i.e. $X$ has a strictly positive ...
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Matrix form for the expected value of discrete variables [duplicate]

I have an expected value: $$ {\rm E}[\log(1+w_1x_1 + w_2x_2)] $$ where $w_1$, $w_2$ are constants and $x_1$, $x_2$ are two discrete variables. The discrete variables $x_1$ and $x_2$ take 2 possible ...
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What order of power mean best estimates the median of a gamma distribution?

Suppose we have a gamma-distributed random variable $X$ whose shape/scale parameters are known to be $\alpha$ and $\beta$. What order $p$ for the sample power mean $\hat M_p[X]$ minimizes $$ (\mathcal{...
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why for stochastic gradient one use expectation value?

Why in order to see how far the gradient samples can be far from the true gradient, they use expectation value. For example $$E[\|\nabla f_i(w)-\nabla f(w)\|^2]=E[\|\nabla f_i(w)\|^2]-\|\nabla f(w)\|\...
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How do you calculate the expected value of a discrete distribution without replacement?

Say I have a set of 10 values I want to draw 3 values from, uniformly, without replacement. For instance: $$S = \{0,0,0,0,22.95,0,0,0,19.125,25.5\}$$ With replacement, this seems simple, you just add ...
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Conservative coverage probability when an biased estimator is used for the variance

Suppose that $X_n\sim N(\mu, \sigma^2_n)$. Thus, to construct a 95% CI for $\mu$, we can use $X_n\pm 1.96 \sigma_n$. The coverage probability, $P(\mu\in [X_n\pm 1.96 \sigma_n])$, is equal to 95%. ...
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Proving the expectation of a variable in a stochastic process

Problem Information packets arrive at a server with a poisson process having rate $\lambda = 2$ per hour. The server processing time for a packet follows the distribution : $f(x) = 1, 0\leq x\leq1$ ...
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Condition expectation of $X_1|\bar{X}$ [duplicate]

I've just learnt about conditional expectations and I'm confused about how we evaluate $E[X_1|\bar{X}]$ where $X_i\sim N(\theta,1),1\le i\le n$ and hence, $\bar{X}\sim N(\theta,1/n)$. Can someone ...
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Least square estimate expected value and variance of linear model

I am practice some exercises. Here it goes. "Assume we fit the simple model \begin{equation} \hskip 5cmy=X_1\beta_1+\epsilon \hskip 5cm (1) \end{equation} however the true model is \begin{...
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3 votes
1 answer
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Interval for the expected value

X is a Random Variable that can take values only in $[0,10]$. Suppose $P[X>5] \leq \frac{2}{5}$ and $P[X<1]\leq \frac{1}{2}$. Then what is the interval for the $E[X]$? Answer : $E[X]\geq0.5$ and ...
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5 votes
1 answer
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Do the principal values of higher moments exist for the Cauchy distribution?

It is known that the Cauchy distribution has undefined moments, and that the expectation has a principal Cauchy value $\operatorname{PV}\left( \mathbb{E} [X] \right)$ of zero. I wonder if $\...
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conditional quantile and conditional expectation

I was reading some papers and I found some parts are tricky to understand. Assume I have price data , what does it mean to calculate the conditional mean of the price data given yesterday price ? ...
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Expected Value of $x_1 \exp(a_1 x_1 + a_2 x_2 ... a_n x_n)$ when X is multivariate $N(0, \Sigma)$ [closed]

Which is the $E(x_1 \exp(a_1 x_1 + a_2 x_2 + \dotsm + a_n x_n)$ when X is an n-random vector distributed multivariate normal (0, $\Sigma$).
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Analogous result to Isserlis' theorem for mixed absolute product-moments of multivariate normal distribution

Suppose that $(X_1, \cdots, X_n)$ have a joint normal distribution. If $n = 2m + 1$, then $\mathbb{E} \left[ \prod_{j=1}^n X_j \right] = 0$. This can be argued from the symmetry of the multivariate ...
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The P-value as the expectation of an indicator function

I am familiar with the notion that $Pr(A) = \mathbb{E}[1_{\omega \in A}(\omega)]$, given some suitable measure theoretic assumptions. I seem to recall a comment on a ...
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mean and covarince matrix of AR(1) [closed]

assume I have a price data called pt, I fitted AR(1) model p_t= alpha + beta pt_1 + e_t , ...
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1 answer
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Show Cox regression estimating equation is unbiased

Let $N(x) = I(X \leq x, \Delta=1)$ be the counting process for observed failure events, where $X = \min\{T,C\}$ and $\Delta = I(T \leq C)$, for censoring time $C$ and failure time $T$. Assume that $T$ ...
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Issue with Casella&Berger derivation of EM likelihood equality

In the explanation of the EM (Expectation maximization) algorithm p.328 in the book "Statistical inference" by G. Casella and R. Berger, 2nd edition, they present the following: $\mathbf{Y} =...
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Expectation of a random variable probabilistically defined in terms of other random variables

For instance, $$X = \begin{cases} Y, & \text{with probability} ~0.3,\\ Z, & \text{with probability} ~0.7, \end{cases}$$ where $Y$ and $Z$ are random variables with known distributions. How ...
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Suppose $X, Y, Z$ are random variables. By Tower rule, $E(X) = E(E(X|Y))$. Is $E(X) = E(E(X|Y, Z))$?

Suppose $X, Y, Z$ are random variables. By Tower rule (iterated expectations), $$E(X) = E(E(X|Y))$$ My question is, is $$E(X) = E(E(X|Y, Z))?$$ My attempt (assuming $X, Y, Z$ are discrete r.v.): \...
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Show that, for any real numbers a and b such that m ≤ a ≤ b or m ≥ a ≥ b, E|Y − a| ≤ E|Y − b| ,where Y be a random variable with finite expectation

Let $Y$ be a random variable with finite expectation, and $m$ be a median of $Y;$ i.e., $P(Y \le m) \ge 1/2$ and $P(Y \ge m) \ge 1/2.$ Show that, for any real numbers $a$ and $b$ such that $m\le a \le ...
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4 votes
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Wainwright's HDS Exercise 5.5 Gaussian and Rademacher Complexities

This is about upper bounding Rademacher complexity by Gaussian complexity but I am only asking about a step in the proof and the actual question is not so important. A similar question was asked ...
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1 vote
1 answer
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Expectation and Variance of Moran's I under the Null

Moran's I is a statistic used to measure spatial autocorrelation. For a set of $N$ spatial units where we get measurements $\mathbf{x} = (x_1, x_2, \cdots, x_N)^T$, and a weight matrix between the ...
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some thought about independence and orthogonal, please comment on this if it's wrong

It seems that linearly independent is totally different from independent of random variable concept. Non-zero vectors Orthogonality must imply linearly independence. In Statistics, the relation of ...
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