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 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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212 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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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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Holder's inequality [migrated]

Given random variables $X$ and $Y$, Holder's inequality states that: \begin{equation} ||XY||_1 \leq ||X||_p ||Y||_q , \end{equation} for $\frac{1}{p} + \frac{1}{q} = 1$, and $p,q \in [1, \infty]$. ...
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73 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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23 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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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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480 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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42 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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47 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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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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Waiting time for successive occurrences of a result, when rolling a dice

In consecutive throws of an ordinary dice, 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 ...
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30 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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51 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: $$ E(y_1)=\alpha $$ $$ ...
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Interpretation of Count Models Based on Weibull Interarrival Times?

This question is an extension of this question, but more specific. This paper E. Bradlow et al is a Weibull counting model which I am using to estimate how many failures will happen between ...
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22 views

Relation between mean of the hypergeometric distribution and binomial

The mean of the hypergeometric distribution is: $n \frac{K}{N}$ where: $n$ is the number of draws $K$ is the number of successes N is the size of the finite population. As the population size, ...
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16 views

Maximizing expectations vs Mode maximization

In many statistical problems, I see the following formulation for maximizing rewards: Assuming that my total reward $R$ is the sum of individual rewards $R$: ...
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63 views

Expectation / minimizing variance of weighted sum of means

(I'm assuming the second $\bar x$ should be a $\bar y$), but I'm mostly confused how to solve this problem because it seems like since $\bar x$ and $\bar y$ are values, not random variables, $W$ is ...
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17 views

Conditional distribution of Inverse Wishart

Suppose $\begin{bmatrix} K_{11} K_{12}\\K_{12}^T K_{22} \end{bmatrix}\sim\mathcal{IW}\left(\eta,\begin{bmatrix} \Sigma_{11} \Sigma_{12}\\\Sigma_{12}^T \Sigma_{22} \end{bmatrix}\right)$. What is the ...
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1answer
26 views

How to compare observed and expected outcomes for continuous data

I am working on some data, more specifically some predictions of some outcomes. The predictions vary on the continuous scale, between $-3$ and $3$. They can for example be: $x_1=-2.4, x_2=-2.1, ...
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16 views

Intuition behind Expected Value of conversion events

I'm trying to develop a high level model to value events in a marketing conversion funnel. To take a simple e-commerce example: You start with leads in the form of ad clicks. Some % of these ad ...
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101 views

E[g(Y)] proof question

This is one of the theorems in my stats text, and I need some help understanding the proof. How can the summand($g_{i}$) be out of its summation sign when multiplying? I thought you can never ...
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33 views

Finding the MSE of the MLE of N?

If X is taken from a uniform discrete distribution, how can I find the MSE of the MLE of N? (I got disturbing and contradictory answers). I am having some trouble with calculating the mean squared ...
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135 views

how can calculate $E(X^Y|X+Y=1)$

let $X,Y$ are two independent random variables Bernoulli with probability $P$. how can calculate $E(X^Y|X+Y=1)$
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Proving increasing function using expectation

Given $c>0$ and define a function $f:(0,\infty)\mapsto \mathbb{R}$ by \begin{equation} f(\sigma)=\frac1{\sqrt {2\pi}\sigma}\int_{-\infty}^\infty\max\{-c,\min\{x,c\}\}^2e^{-\frac{x^2}{2\sigma^2}}. ...
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Does computing this confidence interval make sense here?

I work in a medical company and we often analyze the days patients stay in hospital longer than needed. For example we have these data: ...
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49 views

Expectation of density ratio of two iid variables

Let $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent RVs and let $f$ be their density function. I'd like to compute the expectation of the density ratio \begin{align} ...
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40 views

Transform moments of a random variable to fit the moments of another

Let's say we have the a set $(X_i,Y_i)$, $i\in I$, $I$ is an arbitray finite set of indexes, and the model $$ Y = g(X\beta)$$ Using some method, we obtain the individual first four moments of the ...
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260 views

Expected value of sample median given the sample mean

Let $Y$ denote the median and let $\bar{X}$ denote the mean, of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
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Chi squared test with expected frequencies coming from another observation

I have done an experiment to test the performance of a system. In this experiment I collected answers from people. These answers are categorical (they were able to select on a 5 points scale). Each ...
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Expected value of maximum likelihood coin parameter estimate

Suppose I have a coin toss experiment in which I want to calculate the maximum likelihood estimate of the coin parameter $p$ when tossing the coin $n$ times. After calculating the derivative of the ...
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Consistent, non-parametric, robust (to fat tails) estimation of expected value of an asymmetric distribution

Question: Is anyone aware of a consistent, non-parametric estimator of the expected value of an asymmetric distribution that is robust to fat tails? What if we constrain ourselves to the class of ...
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68 views

What is the expected value E(X), E(Y), Var (X,Y)

Suppose there are three cards labeled with values 1, 2, and 3, and two of them are chosen at random (without replacement). Let the random variable $X$ be the value on the first card chosen, and let ...
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Expected value of coin tossed twice

Here's the problem: A (Laplace) coin with sides $0$ and $1$ is tossed twice, each time independently from the other. Let $f:=$ maximum of both results $g:=$ sum of both results. What is the ...
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Martingale and deterministic functions

Suppose: $u_t \sim N(0,1) \ iid.$, $X_t = g(X_{t-1}) \cdot u_t$ whereas $g(X)$ can be any deterministic function. Is this sufficient to define a martingale? So does it hold: $E(X_t|X_{t-1}, \ldots , ...
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35 views

I cannot calculate expected value

I cannot understand this answer , why we get probability of x=2,3?? - the question tell us that the play end when team won 2 games only not 3 - and why EX[x] is maximized at (d^2 E(x))/dp^2 i need ...
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31 views

Expectation of a function of two variables, not with respect to the joint distribution

I am confused by the following expectation which appears within equation (1) of the following paper http://statweb.stanford.edu/~jhf/ftp/trebst.pdf: $E_{X}[E_{Y}[L(Y,F(X))]|X]$ I am confused by the ...
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83 views

how calculate expected value

(Ross [2009], p.162) The current in a semiconductor diode is often measured by the Shockley equation I = I0(e^aV-1) where V is the voltage across the diode; I0 is the reverse current; a is a constant; ...
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49 views

What is the expected partial value function really called?

If f is a pdf, the integral of x*f(x) over the entire range where f(x) > 0 gives, of course, the expected value. Suppose that integrate the same function, x*f(x) from negative infinity up to t, ...
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73 views

Integrating Gaussian white noise over a Gaussian density

I have encountered the following integral: $$ \int_{- \infty}^{+ \infty} X(\theta) \frac{\exp \big \{-\frac{\theta^2}{2\sigma^2} \big \}}{\sigma\sqrt{2 \pi}}d\theta $$ where, for fixed $\theta = ...
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292 views

Expectation of reciprocal of a variable

I am confused in applying expectation in denominator. E(1/X)=? can It be 1/E(X)?
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62 views

Variance of sample Variance

In page 292 of Introduction to Mathematical Statistics by Hogg and Craig it is stated that in order for the variance of the sample variance, i.e. $\text{var}(S^2)$ to exist we need to assume that ...
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65 views

Conditional expectations by conditioning on functions of random variables

I have conjectured the following: Let $f:\mathbb{R}\supseteq A \rightarrow B \subseteq \mathbb{R}$ be an injective function. Let $X$ be a random variable with support $A$ and $Y$ be some random ...
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I want to show $E(B(t)-B(s))^4=3(t-s)^2$

Let $B(t)$ and $B(s)$ are brownian-motion I want to show $$E(B(t)-B(s))^4=3(t-s)^2$$ thanks for help.
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Relationship between expected score and expected rank in a round robin tournament

Suppose you have $n$ players play a round robin tournament. A win adds 1 to the winning score of a player while a loss adds nothing. After the round robin tournament is finished the players are ...
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Is expectation of univariate variable dependent on the multivariate

Suppose I have a mutivariate Gaussian distributed variable $u\sim\mathcal{N}(\mu,\Sigma)$, where $\Sigma$ is a dense matrix. I wish to calculate the expectation of $f(u_i)$. Is $$E(f(u_i))=\int ...
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Is there an alternative representation for $E[\max\{X,Y\}]$?

Suppose we have $X,Y$ i.i.d. Is there a simplified form of $E[\max\{X,Y\}]$? Is it just $\max\{E[X],E[Y]\}$? That doesn't seem right, because the latter would just be $E[X]$, and it seems like taking ...
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170 views

Show that if $X\ge 0$ , $E(X)\le \sum_{n=0}^{\infty}P(X>n)$

If $X$ is a random variable and also let $X\ge 0$. I want to show $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.
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153 views

Expected Value of cumulative distribution function

Let $\varepsilon$ be a Gaussian distributed random variable with mean $\mu_0$ and standard deviation $\sigma_0$. Is it possible to compute/approximate the expected value $$ \begin{eqnarray} & ...
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420 views

Expected value of minimum order statistic from a normal sample

UPDATE Jan 10th 2014: the mistake was found - a math typo in one of the sources used. Preparing correction... UPDATE Jan 25th 2014: the mistake is now corrected. Please ignore the calculated values ...