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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3
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2answers
42 views

WLLN: can expectation exist but be infinite?

WLLN: Let $\{h_i, i = 1, \dots n\}$ be an $m \times q$ sequence of iid random variables with mean $\mu = E[h_i]$ that exists and is finite. Then $1/n \sum_{i = 1}^n h_i \rightarrow \mu$ in ...
0
votes
0answers
25 views

Is it always true that $E[E[X|Y]^2] = E[X|Y]^2$? [duplicate]

X and Y are random variables. So $E[X|Y]$ is conditioned on a random variable. Do we always have: $$E[E[X|Y]^2] = E[X|Y]^2.$$ I have the doubt because I know that $E[X|Y]$ is a random variable ...
1
vote
0answers
48 views

On $E[E[Y|X]|X]= E[Y|X]$

I am trying to simplify $E[YE[Y|X]|X]$ can I use this property: $$E[E[Y|X]|X]= E[Y|X]$$ If yes I have never seen a Proof of this property (that seems very reasonable), could I have a reference? If ...
1
vote
0answers
28 views

on the minimization of: $E[((Y-f(X))^2|X]$ [duplicate]

I am having troubles solving this exercise: Deduce that the random variable $f(X)$ that minimizes $E[((Y-f(X))^2|X]$ is $$f(X)= E[X|Y]. $$ I proceeded in this way: $$E[(Y-f(X) + E[Y|X] - ...
4
votes
2answers
33 views

Expectation of von Mises Fisher Distribution

The von Mises- Fisher distribution is defined as $$ \frac{\kappa^{p/2-1}}{2\pi I_{p/2-1}(\kappa)}\exp(\kappa \mu^Tx) $$ It is defined over the unit sphere i.e. $||x||_2^2=1$. My question is what is ...
0
votes
0answers
23 views

Expected value of a squared fraction of Y

I need to work out the following: $$ E[(\frac {Y(x+h)-Y(x)}h)^2] $$ I've already worked out the below and am supposed to use it to work out the above. $$E[(Y(x+h)-Y(x))^2]$$ I'm not able to find ...
2
votes
1answer
22 views

Chi-Squared Goodness of Fit Test Alternative? - Zero Can't Be in Denominator

I have 5 zones(categories) in which a certain percentage of total sinkholes exist. I have 5 different maps that I am testing to see which one provides me with the best fit to my expected percentages ...
2
votes
2answers
63 views

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 ...
5
votes
1answer
122 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 ...
3
votes
0answers
77 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 ...
0
votes
1answer
98 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 $$\epsilon_i=\mu(t_i-t_{i-1})-\sum^{i-1}_{k=1}\frac{\alpha}{\beta}\left(1-e^{-\beta(t_i-t_k)}\right)$$ ...
0
votes
0answers
42 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 ...
2
votes
1answer
70 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 ...
0
votes
1answer
66 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} = ...
1
vote
2answers
67 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 $\text{E}\left((x - \bar{x})(y - \bar{y})\right)$ ...
3
votes
4answers
42 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) = ...
1
vote
1answer
41 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 ...
6
votes
2answers
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 ...
4
votes
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 ...
2
votes
2answers
73 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: ...
0
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0answers
21 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 ...
2
votes
1answer
55 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 ...
2
votes
2answers
94 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 ...
0
votes
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 ...
0
votes
1answer
65 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 ...
7
votes
1answer
72 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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. ...
1
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0answers
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) ...
8
votes
2answers
207 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 ...
1
vote
1answer
50 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 ...
5
votes
3answers
306 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 ...
4
votes
0answers
116 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 ...
0
votes
0answers
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 ...
0
votes
1answer
50 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 ...
0
votes
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. ...
0
votes
0answers
30 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 ...
4
votes
1answer
52 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
votes
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: $$E[B_1]=\int_0^{\infty}E[B_1 \vert Y_1 = y] dG(y)$$
3
votes
1answer
101 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 ...
1
vote
0answers
84 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
votes
1answer
61 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 $ ...
3
votes
2answers
99 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 ...
0
votes
1answer
56 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 ...
1
vote
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 $S_{I}$, ...
1
vote
1answer
248 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 ...
0
votes
2answers
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 ...
2
votes
1answer
136 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 $1/\mu$, ...
0
votes
0answers
81 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 ...
1
vote
0answers
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 ...