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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25 views

Expected value of least squares estimator $\hat{\beta}$

Given $\hat{\beta} = (X^{T}X)^{-1}(X^{T}Y)$, how do you derive the expected value? I found answers for finding the variance matrix but not the expected value. Thank you kindly.
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1answer
16 views

MSE decomposition to Variance an Bias Square

In showing that MSE can be decomposed into variance plus the square of Bias, the proof in wikipedia has a step, highlighted in the picture. How does this work? How is the expectation pushed in to the ...
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31 views

Bound on the expectancy of the maximum level in skip list

Let $M$ be a random variable for the maximum level of skip list, $M$ is a positive integer, $k$ is an integer from 0 to $\infty$, and $$ \Pr(M>k) = 1 - (1-p^k)^n \leq np^k $$ In the article Skip ...
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38 views

Probability that uniformly distributed points in a square region form a cluster

I have a known number of points N uniformly distributed in a square and I want to solve the expected number of clusters of points. I cluster is formed by a growing algorithm. Starting at a point p, ...
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28 views

Improper use of an expectation?

A derivation in a paper (theoretical ecology--there are often mathematical errors there) I am reading essentially uses the following line: $\frac{1}{n}\sum_{i=1}^{n}X_{i}=E\left[X_{i}\right]$. This ...
2
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40 views

Definition of Expectation clarification

In Econometric Theory of Davidson (2004) I read (p. 446): ''' In terms of the parent probability space $(\Omega, \mathcal{F}, P)$ this implies a partition of $\Omega$ into sets $A_1, \ldots, A_n$, ...
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12 views

Correlation co-efficient calculation

Suppose an experiment having $r$ possible outcomes $1,2,\dots,r$ that occur with probabilities $p_1,p_2,\dots,p_r$ is repeated $n$ times independently. Let $X$ be the number of times the first outcome ...
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18 views

I don't understand the solution to this Chernoff inequality?

I have a sum, $S_n = \sum_{i=1}^n X_i$ of n iid Poisson distributed random variables $X_1,...X_n$ I am supposed to apply the Chernoff bound to $S_n$. My professor gave us the solution: However, I ...
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12 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
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1answer
16 views

Expected value of a dice game

Say that I have a dice game. You can roll the die first and then have two choices. First, take the dollar amount of the number that shows up (if you rolled a 5, you get $5 ). Second, you can ...
4
votes
2answers
78 views

Expectation of $(X + Y)^2$ where $X$ and $Y$ are independent Poisson random variables

I would really appreciate anyone's help with this problem: (let $E$ denote expectation) Suppose $X$ and $Y$ are independent Poisson random variables, each with mean $1$. Find: $E[(X + ...
3
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1answer
61 views

You will randomly select 10 balls from the box with replacement what is $E(\bar X)$

A box contains 100 numbered balls - 21 with the number 1, 36 with the number 2 and 43 with the number 3. You will randomly select 10 balls from the box with replacement and you take the mean of the ...
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1answer
51 views

Expected lifetime of a device with two parts each having spares?

Consider a device with two parts : (1) and (2). Part (1) has 2 spares and part (2) has one spare. Lifetime of part (1) and its spares have iid exponential distribution with rate lambda. Lifetime of ...
2
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0answers
31 views

Expected numbers of distinct colors when drawing without replacement

Consider an urn containing $N$ balls of $P$ different colors, with $p_i$ being the proportion of balls of color $i$ among the $N$ balls ($\sum_i p_i = 1$). I draw $n \leq N$ balls from the urn without ...
3
votes
1answer
57 views

Can someone provide an proof for $E[P[A|X]] = P[A]$

I'm tired of seeing the word "trivial" for this equality on every single lecture notes I could find online. Can someone please show me why this is indeed trivial? Thank you!
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1answer
47 views

Variance of product of two random variables

I’m trying to calculate the variance of a function of two discrete independent functions. The first function, “f(x)”, returns a value of 0 with probability 0.243, a value of 1 with probability 0.306, ...
2
votes
1answer
39 views

Calculating the expected value and variance of an estimator of a normal quantile

I don't quite understand how to use the estimator function and the variance function and plug in the sample mean. I expected that we would plug in the value $\bar X - 1.645s$ into $E(s)$ and $V(s)$. ...
2
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1answer
52 views

Proving for an AR(2) process that $E[X_t | F_{t-1}]=E[X_t | F_{t-2}]=E[X_t | F_{t-3}]$

An exercise states: Using the law of iterated expectations applied to an AR(2) process, verify that $E_{t−k} . . . E_{t−1} (X_t ) = E(X_t |F_{t−k} ) $ for $ k = 1, 2, 3 $ where $ E_{t−k} (X_t ) = ...
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0answers
32 views

Expected value non-independent random variables

Let $X$ be a set of costumers, {$x_1, ..., x_N$}, each $x_i \in X$ have a discount $p_i$ in the interval $[0,1]$, it means if $p_i$ is 0.3, $x_i$ will pay only 0.3 of the entire value. I want to know ...
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0answers
12 views

Estimating the loss between two Beta distributions

Suppose I have two coins, $A, B$ that each come up heads with probability $p_A, p_B$. Starting with a uniform prior on the values of $p_A, p_B$, and seeing data $s_A$ heads out of $N_A$ attempts, ...
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0answers
33 views

Expected value of a function that is not sampled uniformly

How can I calculate the expected value of a random variable $R(\Omega)$, when the samples are not i.i.d? In my specific case, I have more samples at lower values of the parameter of the function, ...
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0answers
28 views

Loss Elimination Ratio

The values in Table 8.2 are available for a random variable X. There is a deductible of 15,000 per loss and no policy limit. Determine the expected cost per payment using X and then assuming 50% ...
2
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1answer
67 views

Expected value of dot product between a random unit vector in $\mathbb{R}^n$ and another given unit vector

I am wondering what is the $\mathbb{E}[(x\cdot v)^2]$ where $x$ is a random unit vector in $\mathbb{R}^n$ and $v$ is a given unit vector in $\mathbb{R}^n$. By $(x\cdot v)$ I mean the dot product ...
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69 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 ...
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26 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 ...
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54 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 ...
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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] - ...
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42 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 ...
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30 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
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1answer
56 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 ...
3
votes
2answers
71 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 ...
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1answer
140 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
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0answers
78 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
324 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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44 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
86 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
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1answer
77 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
80 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
43 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
44 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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votes
2answers
92 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
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0answers
87 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
88 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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28 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
66 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
101 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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votes
1answer
89 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
67 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
88 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
26 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 ...