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 value and variance for the changed range

I have a distribution with E[X]=-8 and Var[X]=60 and the range of the data [-256,255]. I want to convert this distribution of range [0,256] taking the absolute value of data set. What will be the ...
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Loss Size Index Function of A Lognormal Random Variable

I have this tutorial question and I've gone through the solutions, getting all but one line of working. I broke down the question to this point but I can't seem to get out the following. So Loss Size ...
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1answer
37 views

Convergence of Bernoulli sampling procedure

Let $X$ be a variable which has density $f_{X}$ and mean $E[X]=\mu$ Define $B$, a Bernoulli variable, which has a probability $P(b_{i}=1)=p_{i}=\frac{1}{N^\gamma}$ Consider the following ...
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12 views

Homogeneity of multivariate k:th cumulant

I want to show that $c_k[AXB]=(B'\otimes A)c_k[X](B\otimes A')^{\otimes k-1}$ for a random matrix $X$. This is how I started but I am not able to finish: Characteristic function ...
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17 views

Deriving expectation involving Wishart distributions $E[\bf{A(A'WA)^-A'W}]=\bf{A(A'\Sigma A)^-A'\Sigma}$

I have a problem deriving two expectation involving Wishart distributions with mean zero. Let $\bf{W} \sim {W_p}({\bf{\Sigma }},$$n\bf{)}$ and $\bf{A}$$: p\times q$. Prove that ...
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24 views

Expected value of bounded function?

The expected value of a function is $$E[g(x)] = \int_{-\infty}^{\infty}g(x)f(x)dx.$$ What happens if $g$ is a function such as $g:\mathbb{R}\rightarrow]a,b[$? Does the expected value exist? Should ...
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30 views

Does conditional expectation represent the times that something happens?

Suppose that five coins are each tossed until the first head is obtained on each coin and where each coin has a probability $\theta$ of producing a head. If you are told that the total number of tails ...
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36 views

Existence of expected value of function

According to Wikipedia, the expected value of a continuous random variable is $$E[X] = \int_{-\infty}^{\infty} xf(x) \mathrm{d}x.$$ Suppose $f$ is a function such as $f:\mathbb{R}\rightarrow ...
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22 views

Express $E(x^{\alpha})$ in terms of $E(e^{-\zeta x})$? to a 1st or second order?

I have a random variable, $X$, and am able to find $\mathbf{E}(e^{-\zeta X})$ for many $\zeta$ (through the Laplace transform solving an ODE as this actually evolves over time) Is there any way I can ...
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47 views

Cumulative count of events

Consider an active chat room which has less than $N$ messages so far, each with a timestamp. How can I determine the first day where the expected cumulative message count has a $k$ chance of ...
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62 views

Is it possible for a random variable with 0 mean to have a positive expectation after compounding many observations?

Basic example, assume that the expected daily return of the S&P 500 stock market index is 0, i.e., the return on any given day of the stock market is 0.0%. But, we also (generally) expect that ...
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93 views

Expected value of x in a normal distribution, GIVEN that it is below a certain value

Just wondering if it is possible to find the Expected value of x if it is normally distributed, given that is below a certain value (for example, below the mean value). Thanks,
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180 views

Expected value as a function of quantiles?

I was wondering where there is a general formula to relate the expected value of a continuous random variable as a function of the quantiles of the same r.v. The expected value of r.v. $X$ is defined ...
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1answer
53 views

Shrinkage of the Sample Covariance matrix

Assume we have N independent and identically distributed random vectors $X_1, X_2, ..., X_N$ where each of them is of size p $\times$ 1. The sample covariance matrix, denoted here by $S$, is computed ...
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57 views

How to advise using probabilty [closed]

A game of chance entails three coins. For each head that comes up, the player receives £10 and loses £8 for any tail that comes up. To play the game, one pays £15. Advise a player whether it is ...
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14 views

conditional/unconditional expectation and variance for an AR(1) process

We have an AR(1) process, $X_t=\alpha X_{t-1}+\varepsilon_t$ with $\varepsilon\sim(0,\sigma^2)$, $X_0=0$ and $|\phi|<1$. We have the conditional expected a value with respect to $X_{t-1}$: ...
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64 views

Expected value of product of non independent Bernoulli random variables (correlations are known)

I've asked a question about getting the joint probability distribution for $N$ Bernoulli random variables, given the expected value for each one ($E[X_i]=p_i)$ and it's correlations ...
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40 views

Working with expected value of a matrix

If I have two matrices $C$ and $D$ of the same size. If I know that the expected value of $C$, denoted by $E(C)$, is equal to $D$. So $E(C)=D$. In this case, $E(diag(C))$ will be equal to $diag(D)$, ...
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5 views

Find the expected frequency of some state in a state sequence of length N given a transition matrix M

I can represent stochastically-articulated sequences of states using a transition matrix M where a given entry in cell (i,j) corresponds to the probability of state j given that the current (or, most ...
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127 views

What is the expected value of $\frac{X}{X+Y}$?

I am trying to find the expected value of $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg]$. I started with writing $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg] = ...
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105 views

Expected value of tangent of a normal random variable

If $z\sim N(\mu,\sigma^2)$ What is $E[\tan(z)]$ and $E[\tan^2(z)]$? Generally, it seems that the expectation does not exist. How about if $z$ is bounded $(0,\pi/2)$? Update: Theoretically, my ...
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CLT and 2 variables

Okay so there are 2 variables $D_i$ and $V_i$. Now $D= D_1 + D_2 + ... + D_N$ and $V = V_1 +.. +V_N$ Now I know the relationship is such that $E[D_i - a*V_i] = 0$ and $Var[D_i - a*V_i] = E[D_i]$ ...
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27 views

What is expected count formula for zero-inflated negative binomial regression?

My IT department wants me to translate my zero-inflated negative binomial regression model into a formula for calculating expected count which they can hard code into SQL. I'm running the model in ...
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31 views

Given a set of numbers from 0 to 100, what's the chance that the next number will be less than 10?

I performed a set of trials. Every trial returned a number 0<=N<100. What's the chance that in the next trial, the picked number will be less than 10?
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59 views

When do Taylor series approximations to expectations of (entire) functions converge?

Take an expectation of the form $E(f(X))$ for some univariate random variable $X$ and an entire function $f(\cdot)$ (i.e., the interval of convergence is the whole real line) I have a moment ...
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61 views

Getting Expected Value from Monte Carlo Simulation

There are two independent uniform continuous random variables $X$ and $Y$ (such that $0 \leq X \leq 10$, $0 \leq Y \leq 10$). The function $f$ is the difference between the two random variables ...
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28 views

Double expected value, which comes first?

In the following equation, the outer expectation is over the distribution $X_i|T_i = 1$ $\tau|_{T = 1} = E(E(Y_i|X_i, T_i = 1) - E(Y_i|X_i, T_i = 0)|T_i =1)$ Are we taking the expected value of ...
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5 views

How do I find the expected coverage of the domain after K ranges are selected?

If I have a domain of 0 to 1000000 (but I doubt the domain matters, just that the ranges can overlap). I then chose 1% ranges uniformly, allowing for ranges to overlap completely or partially. How do ...
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47 views

How to estimate a chance of getting in the first positions, given previous tries?

I perform a series of experiments that return a number N that is between 1 and MAX. MAX varies between experiments. For example: ...
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52 views

$E_{\theta_1}[\ell(\theta_2;X)] $

I am faced with the following in my "Statistical inference book" $E_{\theta_1}[\ell(\theta_2;X)] $ where $\ell(\theta_2;X)$ (loglikelihood) is $\log [P(\theta_2;X)]$, X is a random variable. What ...
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47 views

Expected value of a diagonal

If I have $E[A] = B$, where $E$ is the expected value, $A$ and $B$ are square matrices and $\text{diag}$ refers to the vector of coefficients on its diagonal. In this case, what will be the value of ...
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29 views

Non significant A/B test, what is the future expected value?

Suppose you run an A/B test with a result that is not significant. For example: Version A: 10,000 views, 100 clicks Version B: 10,000 views, 115 clicks What is ...
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27 views

Expectation over training and testing set

If I have a linear regression model with $p$ parameters that fit by least squares a training set $(x_1, y_1),...,(x_n, y_n)$ drawn at random from a population, and we have some test data ...
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1answer
68 views

Computation of expected values

I have a $N$-dimensional (normal) random vector $\mathbf{X}$, where $N$ is large, and a function $f : \mathbb{R}^N \to \mathbb{R}$. My goal is to compute $\operatorname{E}[f(X)]$ or at least ...
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117 views

Expected number of times you spent in a state of an absorbing markov chain, given the eventual absorbing state

It's well known that, if $Q$ is the matrix of transient state transition probabilities, and $$ N = \sum_{n=0}^{\infty} Q^n = (I - Q)^{-1}$$ then $N_{ij}$ describes the expected number of times the ...
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32 views

How do i show multiplication of covariances?

i am trying to show the following $$Cov(a,b) = \frac {Cov(a,x) * Cov(b,x)} {Var(x)}$$ I am a bit lost on how to expand the numerator term, so i wrote the following $$Cov(a,b) = \frac ...
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32 views

Time to completion

Say a person needs to go from A to B and also needs to complete a task. He may complete the task at either A or B. The task, say, takes 10 minutes. It takes 60 minutes to go from A to B after boarding ...
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Expectation of $b^T \operatorname{sign}(Ab)$

I'm trying to compute the expectation of: $$b^T \operatorname{sign}(Ab)$$ Where $b$ is a $n\times1$ vector of independent Bernoulli random variables: $$P(b_i = 1) = 0.5,\quad P(b_i = -1) = 0.5$$ and ...
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46 views

Shannon entropy and inequality of expectations

Consider two distinct probability distributions $P(X)$ and $Q(Y)$---defined on the same domain---with (Shannon) entropy of $H(X)$ and $H(Y)$. I am interested to prove that $$ H(X) \leq H(Y) \implies ...
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49 views

Using Dirac Delta functions for estimating a probability distribution

I'm having some trouble understanding this slide. It's mentioned in the context of gaussian distributions. I sort of understand the Dirac delta "function". The main difficulty I'm having is with ...
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1answer
47 views

How to compute $P(|X - E_Y[h(y)]| < c)$?

Consider a discrete random variable $Y$, a continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, and ...
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bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \stackrel{iid}{\sim} P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one ...
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31 views

Relationship between expectation and probability with continuous variables

I've tried to word this question to the best of my ability but I may have got some of the terminology wrong. Example: I have a machine producing various lengths of knotted string. If I have a string ...
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135 views

Is it true that $E[Y|W]=E[Y|E(X|W)]$, given that $W$ is $X$ measured with error

In Carroll's 2006 book "Measurement Error in Nonlinear Models", and on p38 it is stated without proof that: [One can] estimate the regression of $Y$ on $(Z, X)$ and then to substitute into this ...
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36 views

Expected number of draws from an urn (without replacement) to get 2 different coloured balls

An urn has $N=18$ black balls and $M=2$ white balls. What is the mean number of balls one must draw to get two different colours. My solution so far is extremely ugly (though effective): ...
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73 views

What is the theoretical difference between using population and sample means for calculating expectation?

In looking at linear combinations of random samples, what is the bridge between substituting the population mean $\mu$ with the sample mean? For instance: Population mean (sample size n): $$E(Y) = ...
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1answer
32 views

What is the meaning of averaging combinations of independent random variables?

I'm getting into the management of multiple independent random variables in determining expectation and variance, but cannot see where averaging of a linear combination of independent random variables ...
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144 views

In which case $\mathbb E[X]=\sum _ix_i P[x_i]$ can be $0$ when all $x$'s are not zero ($0$)?

Say $X$ is a random variable and $x$'s are realizations of $X$ . Say , $\mathbb E[X]=\sum _ix_i P[x_i]=0$ . But I do not understand in which case $\mathbb E[X]=\sum _ix_i P[x_i]$ can be $0$ when all ...
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167 views

Why is $\mathbb E(X)=\sum_{i=1}^{n}x_i P(x_i)$?

If $X$ is a random variable and $x$'s are the realizations form $X$ and $N$ is the population size $n$ is the sample size Which one is correct $\mathbb E(X)=\sum_{i=1}^{N}x_i P(x_i)$ or ...
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39 views

Why moments of expectation are known as “moments” [duplicate]

I am studying moments of expectation, and seen the formulas for computing the moments. There is one thing I am not clear of, and not getting answer for that. Why moments are named as moments? To my ...