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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conditional expectations value

I need to calculate the following integral $$\int_{\mu+c}^{\infty} y\cdot \frac{1}{\sigma\sqrt{2\pi}}e^{(y-\mu-w)^2/2\sigma^2}dy$$ So essentially $y\sim N (\mu+w, \sigma^2)$ and im trying to ...
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1answer
82 views

$\bar{X}$ versus $\mathbb{E}(\bar{X})$?

I was not able to find this question here, so I am going to ask this: What is the difference between $\mathbb{E}(\bar{X})$ (expected value of $X$ bar) and the actual $\bar{X}$? I am very confused ...
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71 views

How do I solve $E\left[ E \left(X|Z \right) E\left( Y|Z \right)\right]$?

I am trying to solve $E\left[ E \left( \mathbf{X}|\mathbf{Z} \right) E \left( \mathbf{Y}|\mathbf{Z} \right) \right]$, (where $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ are random variables) but I am ...
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1answer
47 views

Expected Value of Random Variable

I'm trying to find the expected value of a random variable $t_i$ which is the solution of $$\epsilon_i=\mu(t_i-t_{i-1})-\sum^{i-1}_{k=1}\frac{\alpha}{\beta}\left(1-e^{-\beta(t_i-t_k)}\right)$$ ...
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36 views

Obtain expected shortfall (ES) in value terms from an ES stated in log-returns

Let $r_t^{log} = \ln{\frac{Y_t}{Y_{t-1}}}$ be the log return where $Y_t$ is the portfolio value at time t. If the value at risk (VaR) is defined as minus the 0.05 quantile of the log-returns ...
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69 views

Best statistical notation for expected probability density

Assume that we have two multivariate normal distributions $\mathcal{N}_1 = \mathcal{N}(\mu_1, \Sigma_1)$ and $\mathcal{N}_2 = \mathcal{N}(\mu_2, \Sigma_2)$. We do these two steps: Pick a point, say ...
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63 views

how does the correlation of independent variables affect the correlation of parameters

Suppose $Y$,$X$ and $Z$ are correlated random variables with $N(0,1)$. We have these cross-sectional regression for each time $t$ $Y_{t} = \beta_{t}X_{t}+u_{t} $ $Y_{t} = ...
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2answers
64 views

How to 'read' (understand ) an expected value equation (example inside)

I have just come across expected values and they are giving me a bit of grief trying to understand them. e.g. for covariance the equation is $\text{E}\left((x - \bar{x})(y - \bar{y})\right)$ ...
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4answers
41 views

Conditional variance - $Var(X + U | X) = Var(U)$?

I am wondering if the following equality holds - $Var(X + U | X) = Var(U)$? where $X$ and $U$ are two independent random variables? It seems can we say $Var(X + U | X) = Var(X|X) + Var(U|X) = ...
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1answer
39 views

Monte Carlo computation of expectation when there is dirac delta

Let $Z \sim N(0,1)$ and let $Y=Z$. Suppose I wish to perform the following weird computation: $f(z)=\int f(z|y)f(y)dy=E_Y[f(z|y)]$ and then use Monte Carlo to estimate $E_Y[f(z|y)]$. The problem is ...
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91 views

$E(\frac{1}{1+x^2})$ under a Gaussian

This question is leading on from the following question. http://math.stackexchange.com/questions/360275/e1-1x2-under-a-normal-distribution Basically what is the $E\left(\frac{1}{1+x^2}\right)$ under ...
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0answers
85 views

$E[e^{cX}]$ where $c < 0$ and $X$ is lognormally distributed

I am trying to calculate the expectation $$E[e^{cX}]$$ for arbitrary $c<0$ (for $c>0$ the expectation is infinite) if $X$ is lognormally distributed, i.e. $\log(X) \sim N(\mu, \sigma)$. My idea ...
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2answers
70 views

Finding the Expected Average Distance from the Maximum given a distribution

For a given sample set $S$ with $N$ individual samples $x_i$, I can easily find the average distance from the maximum by doing something like this: ...
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20 views

Expectation of a generalization of Dirichlet distribution

For the standard Dirichlet, the expectation of $X_i$ is $\alpha_i/\alpha_0$, where $\alpha_0 = \sum_i \alpha_i$ (http://en.wikipedia.org/wiki/Dirichlet_distribution). I am considering the following ...
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1answer
51 views

Bounded expectation implied bounded conditional or vice versa?

If $\mathrm{E}\left(X\right)<\infty$ does that imply $\mathrm{E}\left(X|Y\right)<\infty$? How about vice versa? I'm thinking if we condition on an event (say $Y>2$) then if we have ...
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2answers
93 views

Probability that a sum of potential numbers is greater than some value

Say I am about to receive 5 cash prizes and I have the probability of receiving each cash prize. Let's denote a set of cash prizes with $k$. So, below is the set of cash prizes and the set of ...
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1answer
85 views

Drunken cockroach - Trying to meet expected value

Imagine that you have $1000 that you can split however you want. You bet in a cockroach run, but it is not the finish that's interesting. You can bet for the cockroach to go left or right, and you ...
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1answer
60 views

$E(X_1| \overline X ) = \overline X$, the sample mean

Let $X_1, X_2, ..., X_n$ be a random iid sample from a population with mean $\theta$. Now I am wondering about the intuition behind $E(X_1| \overline X ) = \overline X$, the sample mean. If we just ...
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69 views

Distribution of the Rayleigh quotient

For a research project I need to find the expected value of the generalized Rayleigh quotient: $$E\,[w^T A w \ / \ w^T B w].$$ Here A and B are positive definite deterministic p x p covariance ...
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24 views

How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$? [duplicate]

Suppose we have a time series $X_t$ s.t. $X_t \sim^{iid} (0,1)$. How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$? Or, I guess, if ...
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15 views

Expected agreement if random

I am looking to measure agreement between participants in choosing which member of a group is most like certain attributes. I want to calculate the expected agreement if they were to choose by chance. ...
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30 views

Non-Measure Theoretic Argument for Var(X) = 0 iff X is constant (X continuous RV)

I am studying out of DeGroot and Schervish trying to carefully understand the math of prob/stats. In ch 4.3 on variance, they state the theorem that given X a RV whose mean and var exist, then Var(X) ...
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205 views

Percentile Loss Functions

The solution to the problem: $$ \min_{m} \; E[|m-X|] $$ is well known to be the median of $X$, but what does the loss function look like for other percentiles? Ex: the 25th percentile of X is the ...
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1answer
48 views

Variance of sample mean for dependent samples

Suppose I have two discrete independant random variables $X$ and $Y$, and that I'm interested in the expected value of the random variable $W$, where: $$ W= \text{sign}(X-Y). $$ So, W is 1 if ...
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303 views

Let f(x) be some PDF, and F(x) be its CDF. Shouldn't F(x)=.5 give us the expected value of f(x)?

I was playing around in R and have gotten myself very confused about the relation between probability distributions, their expected values, and their cumulative distribution functions. Say we're ...
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115 views

How to find this integral [duplicate]

Let $X_1, \cdots, X_n$ be $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. How to find $E[\Phi(\bar X)]$, where $\bar X:=\frac{\sum_{i=1}^nX_i}{n}$, please? I guess ...
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22 views

Showing that the variance increases with the dimension of the random vector

This is actually related to a more complex question; but I want to re-ask it by trying to simplifying it as possible: 1- We have $n$ dimensional functions of the form $f_n:\mathbb{R}^{n} \mapsto ...
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1answer
49 views

Expectation values of functions

This question is more to do with interpretation than calculation. I have a model which predicts the probability of a detector 'firing' under a certain intensity of signal, or actually in this case ...
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1answer
22 views

Multinomial chi square with small expected values

I'm studying extinction in Austronesian languages, and am trying to find out if a subset of 384 languages is randomly selected with respect to extinction risk from a population of 1249 languages. ...
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28 views

sufficient statistic and KL-divergence: Confusion with an equation

I am reading a paper, which talks about minimising KL-divergence of any arbitrary distribution over a family of exponential distribution. So, given a distribution $p$, we want to compute its ...
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1answer
46 views

Estimating the error in the average of correlated values

tl;dr I can only generate samples of a random variable $X$ using MCMC. How can I find the error in the estimate of the expected value of $X$ based on this MCMC data? The problem I have a "black ...
3
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1answer
79 views

dG(y) in expected value integral

I am wondering what exactly the notation dG(y) inside an integral means, what it's called and where I can read more about it: $$E[B_1]=\int_0^{\infty}E[B_1 \vert Y_1 = y] dG(y)$$
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1answer
92 views

Why is the conditional expectation of prediction error in regression not zero?

The conditional expectation of the error in regression is: $E[Y-X\beta|X=x]$ is not equal to 0. Why is this the case? If you fix all the predictor variables, why does $E[Y]$ - $X\beta$ not equal to ...
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81 views

Practical meaning of expected value, standard deviation & correlation

We've got given annual results of two stock companies described with following values: Company X: expected value $\mu_X=0.05$, standard deviation $\sigma_X=0.02$ Company Y: expected value ...
2
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1answer
60 views

Is it true that if $ \epsilon_t \sim^{iid} (0,1) $, then $ E(\epsilon_t^{2}\epsilon_{t-j}^{2}) = 1 $?

Under the GARCH($m$,$s$) model, it can be shown that $ E(\eta_t\eta_{t-j}) = E[(a_t^{2}-\sigma_t^{2})(a_{t-j}^{2}-\sigma_{t-j}^{2})] = 0 $. In my proof attempt, I came across $ ...
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2answers
92 views

Confused about why we would use expected value instead of MLE when estimating some parameter

I have a conceptual confusion about the use of the expected value of a distribution. Often, we want to estimate the most likely value of something. For example, I have X= ten observations. I know X ...
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1answer
53 views

the Poisson result and Exponential interpretation for spare part requirement analysis

I am confused with with the Poisson result and Exponential interpretation for spare part requirement analysis. I try to calculate the required number of spare parts for a disposable remove-replace ...
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1answer
40 views

Expectation of Truncated & Random Variable

I have what appears to be a relatively simple question, but am struggling to understand how to go about answering it. The general question is as follows: What is the expected value of $S_{I}$, ...
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1answer
247 views

Expected value of a random variable

Random variable $X$ has the probability density function \begin{equation*} f\left( x\right) =\left\{ \begin{array}{ccc} n\left( \frac{x}{\theta }\right) ^{n-1} & , & 0<x\leqslant \theta ...
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58 views

Expected value of multiplication of Identically distibuted random variables

i am trying to understand if the following statement is true: $$ E(XY)= E(X^2)=E(Y^2) $$ if $X$ and $Y$ are identically distributed but not necessarily independent r.v. This means that if the ...
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1answer
116 views

Expected value of inverse?

If I have a random variable $V$ that is normally distributed with some $\mu$ and $\sigma$, then what is the expected value of $1/V$? I tried doing by delta method, and I get expected value $1/\mu$, ...
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71 views

Difference between absolute deviation to population median and sample mean

I have independent variables $X_i\in[0;1]$ and suppose they are uniformly distributed. If you want to minimize the total absolute deviation to a fixed number, how much can you gain from using the ...
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44 views

Testing if alcohol consumption and smoking are independent

The question asks to test if smoking status and level of alcohol consumption are independent using the usual five-step procure at alpha $=0.05$: I am having trouble finding expected values. As the ...
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2answers
577 views

Why maximum likelihood and not expected likelihood?

Why is it so common to obtain maximum likelihood estimates of parameters, but you virtually never hear about expected likelihood parameter estimates (i.e., based on the expected value rather than the ...
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0answers
82 views

Total waiting time at a server with exponential arrival times

Question: In a certain system, a customer must first be served by server 1 and then by server 2. The service time at server $i$ is exponential with rate $\mu_i$, $i = 1, 2$. An arrival finding server ...
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1answer
85 views

In EM derivation why can I sum over the iid variables in the conditional expectation?

In EM when you take the expectation: $E[\log P(y,x \mid \theta)\mid x, \theta']$ $= \sum\limits_yP(y\mid x, \theta') \log P(y,x\mid \theta)$ I understand this but the following part I don't ...
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1answer
62 views

Does A independent of B, and B correlated with C imply that C is independent of A?

Assume 3 random variables, $A, B, C$. If $A \perp B$, but $Cov(B,C) \neq 0$, can we say anything about $Cov(A,C)$? I think it can either be 0 or not 0, but it seems like there should be more general ...
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97 views

Waiting time for successive occurrences of a result, when rolling a die

In consecutive throws of an ordinary die, which of the following two possibilities is more likely to happen first: a) Two successive occurrences of 5 or b) Three successive appearances of numbers ...
2
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1answer
35 views

How many times, in expectation, am I fetching a random sample from a predefined discrete probability distribution of elements?

As part of a simulation that I'm working on, I have a probability distribution over $n$ elements, from which I have to sample a set $S$ of size $m$. That is, each element $e \in S$ must be unique ...
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64 views

In a linear regression, why is the mean of all $y_i$ equal to 0?

I don't understand why the mean for all $y_i$s must equal $0$. What property/properties does the above assertion rely on? I had thought that the mean of $y_i=E(y_i)$, so: $$ E(y_1)=\alpha $$ $$ ...