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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Expected survival time from log-logistic survival model in R from survreg

I'm currently estimating a survival model (accelerated failure time model) with a log-logistic distribution in R using the survival package and the survreg function. I want to simulate expected ...
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1answer
38 views

What is the expected number of coin flips, if you stop when the first coin flip is the same as the last?

In order to calculate the $\text{E}[X]$ where $X$ is the number of total coin flips, this is the approach I took: The probabilities are: $Pr(H) = p$ $Pr(T) = (1-p)$ Define indicator random ...
9
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1answer
97 views

Show that if $X \sim Bin(n, p)$, then $E|X - np| \le \sqrt{npq}.$

Currently stuck on this, I know I should probably use the mean deviation of the binomial distribution but I can't figure it out.
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39 views

Mean of predictive distribution

I observe independent, Poisson-distributed data $ D = \{x_1, ... x_n \} $ with mean parameter $ \mu $. Over $ \mu $ I assume $ Gamma(\alpha_0, \beta_0) $ as a prior (where $ \alpha_0 $ and $ \beta_0 $ ...
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30 views

expectation of a random variable [closed]

Recently a question has made my life difficult. It might be easy to be solved but for me is difficult. Assume X Is a random variable that follows distribution F. Now I am interested in following ...
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16 views

Find expectation or lower bound of log erf

I need to find the expectation of $\log \Phi(x)=\log \left(\int_{-\infty}^x\frac{1}{2\pi}\exp(-\frac{1}{2}s^2)ds\right)$. (I realise this isn't quite the error function, but not sure what to call it). ...
2
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1answer
42 views

Law of iterated expectations with two random variables

Let $X$ and $Y$ be two random variables. I want to calculate $E[X|X<Y]$. I am wondering whether I can use the law of iterated expectations in order to calculate it, i.e. $E[E[X|X<Y,Y]]$. Do I ...
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18 views

If $E[X(t)X(s)]=t \land s $.Show that this process has independent increments

Let $X(t), t\ge0$ be a real-valued Gaussian process with mean zero and covariance function $E[X(t)X(s)]=t \land s $.Show that this process has independent increments.
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1answer
24 views

Why is $E(u^2)=Var(y)$? (Binary Response Model)

I'm trying to show some results in binary response models, and a couple of the proofs use the "fact" that $E(u^2)=Var(y)$, but I can't see why this is. The setup is that $y$ takes on the value $0$ or ...
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2answers
53 views

Can’t Find the Fisher Information of This Function

Can anyone help me find the Fisher information for this function: $$f(x|\lambda) = \lambda\,x^{\lambda-1}\quad \text{ where } \lambda \in [0,1]\,.$$ Thanks in advance!
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25 views

Bounding the expectation of the difference between empirical vs generalization error

Let the (defect) difference between empirical and generalization error be: $$D[f_S] = I_S[f_S] - I[f_S]$$ where the empirical risk is: $$I_S[f_S] = \frac{1}{n}\sum^n_{i=1} V(f_S,z_i)$$ where ...
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28 views

Calculate expected value from discrete beta distribution

I have a lookup table in 2 variables, $Z_l$ and $T_l$. So, $Z_l$ and $T_l$ are vectors with same length where $Z_l$ goes from 0 to 1 and $T_l$ varies between 300 and 2000. If you are curious, $Z_l$ ...
6
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2answers
165 views

Expected value of a product of two compound Poisson processes

I'm working on my master thesis now and I've been struggling with a problem for some while now and no one seems to be able to help me or point me in any direction. So now I reach out to see if someone ...
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26 views

Mean square convergence

I am working on an example in my book and cannot figure out an expectation. Let $$E(T_n)= \frac{n\theta}{n+1}$$ $$E(T_n^2)= \frac{n\theta^2}{n+2}$$ $$g(t)=\frac{nt^{n-1}}{\theta^{-n}}$$ Then ...
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35 views

Finding Expectation of a Random Variable Using Its Joint Marginal Density

If X and Y have joint density function $$ f(x,y) = \frac{1}{y} \,\mathbb{I}_{0<x<y<1}, $$ how do I find the expectation of X or Y? Since E[X] requires us to know the PDF of X, I tried to ...
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38 views

$Y = \beta_0+\beta_1*X+U$ and $W = \gamma_0+\gamma_1*X+\gamma_2*U$, assume $\gamma_2\neq0$. also given is $E(U|X) = E(U)$ . find $ E(U|W,X)$

$Y = \beta_0+\beta_1*X+U$ and $W = \gamma_0+\gamma_1*X+\gamma_2*U$, assume $\gamma_2\neq0$. also given is $E(U|X) = E(U)$ find $ E(U|W,X)$ and conditions under which $E(U|W=w,X=x)$ is an increasing ...
3
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1answer
27 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
20 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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41 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 ...
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41 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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13 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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13 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
18 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
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2answers
82 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
54 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 ...
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53 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
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1answer
58 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
62 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
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1answer
44 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
54 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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33 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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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, ...
0
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35 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
71 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 ...
3
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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 ...
2
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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 ...
1
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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
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2answers
44 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
61 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
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2answers
72 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
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1answer
141 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
80 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
345 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 $$\epsilon_i=\mu(t_i-t_{i-1})-\sum^{i-1}_{k=1}\frac{\alpha}{\beta}\left(1-e^{-\beta(t_i-t_k)}\right)$$ ...