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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Implicit estimator, it's variance and expectaion

Denote with $V_j$ the firm's value of $j$-th firm. We will speak of default, if the firm's value falls below predefined barrier $c \in \mathbb{R}$. The random variable $L_j$ will indicate, if $j$-th ...
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Expectation and variance of a particular estimator [on hold]

I am going to use default data, where $d(t)$ stands for amount of defaults in period $t$ and $t \in \{1,\ldots , s\}$; $n(t)$ is total number of credits. Suppose the following equation holds $$ ...
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319 views

Expected Value and Most Likely Outcome

I was watching this Khan video on Expected Value as a refresher. He mentions in passing that the expected value is the most likely outcome... Well, that's only true because he's using a ...
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exact value runtime for coin flip algorithm

So I have an algorithm like this: inc = 10 for num in Array: if num == 1: return inc else: inc += 1 inc += 1 return inc I want to ...
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63 views

On some manipulations of $E[( X - tY)^2 ]$

I am following some notes and don't fully understand a step: call $0 \le f(t) = E[ (X- tY)^2] $ then by linearity of the expected value we get $$f(t) =E[Y^2]t^2 - 2 E[XY]t + E[X^2]$$ As noted this ...
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Which distribution has an expected product equal to one?

Consider $n$ values $x_{1}$ to $x_{n}$ such that $x_{1}x_{2}\cdots x_{n} = 1$. Importantly, these values are non-independent, as their product must be equal to 1. The trivial case of $x_{i}=1 \quad \...
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Correlation between sine and cosine

Suppose $X$ is uniformly distributed on $[0, 2\pi]$. Let $Y = \sin X$ and $Z = \cos X$. Show that the correlation between $Y$ and $Z$ is zero. It seems I would need to know the standard deviation ...
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Expected value question [duplicate]

Say I am rolling a 20-sided die. What is the expected number of rolls to have rolled each number? I haven't taken statistics in a while so my syntax might be off but this question has been bugging me ...
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Expectations of cosine under von Mises distribution

I'm trying to work out the expectations of a few functions under the von Mises distribution: $ p(\theta \mid \mu, \kappa) = \frac{1}{2\pi I_0(\kappa)} \exp\left\{ \kappa \cos \left( \theta - \mu \...
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The meaning of expected value for discrete random variable in dice experiments

Does the expected value always speak to a payoff? Or can the expected value be thought of independent of payoffs? I don't understand when we say a fair die has an expected value of 3.5. Does that ...
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Truthfulness of statements on the expected values of random variables

Are these statements true or false? Why? $E(|X|)\le 1 + E(X^2)$ $0≤|x|<1+x^2$ for all choices of $x$ with $x$ real number. What with $X$ random variable? if $E(X)<0$ and $ \theta \neq0$ ...
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Monte Carlo method cannot be used

This is an example from notes I don't understand why we should think about the $E(x)$ and $Var(x)$ first?
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computing a marginal

I have tried to solve this exercise Let $X$ and $Y$ be random variables with joint probability density function given by: $f(x,y)=\frac{1}{8}(x^2-y^2)e^{-x}$ if $x>0$ ,$|y|<x$ Calculate $E(X|Y=...
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Expectation of $\frac{X_1^4}{(X_1^2 + \cdots + X_d^2)^2}$

Let $X_1$, $X_2$, $\cdots$, $X_d \sim \mathcal{N}(0, 1)$ and be independent. What is the expectation of $\frac{X_1^4}{(X_1^2 + \cdots + X_d^2)^2}$? It is easy to find $\mathbb{E}(\frac{X_1^2}{X_1^2 +...
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Difference between two methods of random point generation [migrated]

In order to do a monte carlo simulation to estimate expected distance between two random points in $n$ dimensional space I discovered the following two similar looking methods to generate random ...
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68 views

Expectation of a function of a random variable from CDF

Is it possible to calculate the expectation of a function of a random variable with only the the r.v.'s CDF? Say I have a function $g(x)$ that has the property $\int_{-\infty}^{\infty}g(x)dx \leq \...
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Expectation of Maximum of Transformed iid RVs

Is the following reasoning correct? Let $\mathbb{P}(c \leq l) = F_c(l)$ be the CDF of random variable $c$ and $\pi(c)$ a strictly decreasing positive and bounded function where both are continuous ...
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Expectation Value of Standard Deviation from “k” Closest Samples

The question that I'm am trying to answer is how to determine the expectation value of the standard deviation of 'k' closest samples to value 'A'. The standard deviation and mean of the underlying ...
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Is it okay to write the square of expectation of a random variable $X$ as $\mathbb{E}^2(X)$?

Is this notation accepted when I write $\text{Var}(X)=\mathbb{E}(X^2)-\mathbb{E}^2(X)$?
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Expected value of a semi-partial correlation

Say I have 4 random variables. $X^{(1)}$ and $X^{(2)}$ are jointly multivariate normal with mean 0 and covariance $\Sigma_X$, and $Y^{(1)}$ and $Y^{(2)}$ are jointly multivariate normal with mean 0 ...
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Working with percentages of positive variables

Say we run an experiment and observe the following impact on a variable of interest (one row per experimental unit): ...
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Expectation of infinite sum of random wishart matrices

I have problem figuring out how to find the expectation for an infinite sum of random matrices. More explicitly, my problem is: Let $\mathbf{S}_i$ be the maximum likelihood estimator of the sample ...
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Expected Value clarifications using R/Excel

So it has been a little while since I've taken my last statistics course, and I wanted to double check that I am not making any kind of grave errors in my expected value calculations. Quick bit of ...
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Expected number of points in k turns?

I'm not too familiar with expected values so I'm wondering if people could verify if I'm on the right track with my thought process here. If the probability of winning one point in a turn is 1/9, ...
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Minimising MSE of $\sigma^2$ estimator of specific form

I have found a past exam question for a statistics course and can't seem to find the required result. Part A is fine but my working for part B must be incorrect [see below]. Can anyone figure out ...
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How would you calculate $E[\mid x \mid ^{\alpha }], \alpha \in \Re$?

Here $x \sim N(0,1)$. I realize that the expectation won't be defined for $\alpha$ when the integral goes to infinity. I can't seem to figure out which specific values of $\alpha$ would cause this. ...
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Finding the expected value of a continuous random varibale when the commulative distribution is given

I have this distribution function of a random variable X: I wish to find E(X). I have used derivatives to get the density function, compared it to 1, and found that f(t) = (4/5)t+(3/5). I then ...
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Conditional expectation of a univariate Gaussian

Suppose I have a univariate Gaussian distribution with mean $\mu_X$ and standard deviation $\sigma_X$, and I know the random variable $X$ is least some positive value $y$: $X \geq y$. What is the ...
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How to find $E|X+Y|^3$ from related information?

Assume that $$ E(X+Y)=E(X-Y)=0 $$ $$ V(X+Y)=3 $$ $$ V(X-Y)=1 $$ Show that $E|X+Y|\leq\sqrt3$. If in addition, it is given that $(X,Y)$ is bivariate normal, calculate $E|X+Y|^3$. For the 1st part, ...
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Expectation with indicator function

I have the following expectation $$E[x_{t+1} \mathbf{1}_{\{x_{t+1}> z_t\}}]$$ where $x_{t+1}$ is a normally distributed random variable $x_{t+1}\sim N(0,\sigma^2)$, and $\mathbf{1}$ stands for ...
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Expected value of maximum ratio of n iid normal variables

Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma^2)$ and let $X_{(i)}$ denote the $i$'th smallest element from $X_1,...,X_n$. How would one be able to upper bound the expected maximum of the ratio ...
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Expectation of a hazard rate

I need to estimate the expectation of a hazard function, $E[h(x)]$. For instance, for the exponential distribution the result is equal $\lambda$ $E[h(x)]=\int_0^\infty \! h(x)f(x)\mathrm{d}x=\...
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Expected value word problem from Khan Academy

Hello all and thanks for taking the time to read this. I'm closing out the last few sections of statistics in Khan Academy, but there is a problem that is really bugging me. The problem reads like ...
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51 views

Expected value of difference of two order statistics

$X_1,X_2,..,X_n$ is a random sample from the random variable whose pdf is, \begin{align*} f(x)=\lambda e^{-\lambda(x-\mu)},\mu<x<\infty \end{align*} How can we find $E(X_{(2)}-X_{(1)})$, if $n=2?...
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How do we obtain E(exp(C+By-Y'aY))?

How can I prove this ? I did get $B-2uA$ by identifying the $a+by+cy^2= -1/2 (y-u)^2/\sigma^2$ But, I don't know how to get $(2a+\Sigma^{-1})^{-1} \dots$ Am I missing something? A little help would ...
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Expected value of a function of a multinomially distributed random variable

I have a scalar function, $g(x)$, where $x$ is an $n$-vector following a multinomial distribution with mass $f(x;p, N)$, for some probability-vector $p$, such that $\sum p_i=1$ and where $\sum x_i = N$...
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What is the minimum sample size for kaplan meier

I used the "survival" package in R to calculate a Kaplan Meier estimate for survival. An example of my output is like this: ...
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Approximate Order Statistics for lognormal variables

Are there any known formulas that approximate the expected value of the maximum of $N$ i.i.d. lognormal random variables? I am looking for something similar to: Approximate order statistics for ...
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expectation of conditional expectation

Given $(X,Y)$, 2-dimensional probability vector, and let $g: R^2 \rightarrow R, E[g(X,Y)^2 ] < \infty$ and $h:R \rightarrow R, E[h(X)^2] < \infty $, prove the following: $$E[h(X)\{g(X,Y)-E[g(X,...
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Why is the expected value of y written as E(y|x)?

The expected value of the simple linear regression model $y = \beta_0 + \beta_1x + \epsilon$ is typically written as $E(y|x) = \beta_0 + \beta_1x$. Why is it written as $E(y|x)$ instead of just $E(y)$?...
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Expectation of “mixed-variables” obtained from a N(0,1) variable

I have a variable $z$ which is normally distributed with zero mean and unit variance. I should derive $$ E[z^+]$$ $$ E[z^-]$$ $$E[z^+z^+]$$ $$E[z^-z^-]$$ $$E[z^+z^-]$$ where $z^+ =max(z, 0)$ and $z^- =...
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Stationarity of the TGARCH

I'm going through "GARCH models" by Francq and Zakoian (2010). They define the TGARCH(1,1) as $$\sigma_t = \omega + \beta_1 \sigma_{t-1} + \alpha_{1,+}\epsilon_{t-1}^+ - \alpha_{1,-}\epsilon_{t-1}^- $$...
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How to estimate an expected value of f(x,y) when x and y are random

So I have 3 sets of data. I'll call them x, y, and z (it's not a secret or anything what these variables are, I'm just trying not to distract from the question). x has bounds of 0 to 150 and is random ...
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Order Statistics, Expected Value of range, $E(X_{(n)}-X_{(1)})$

$X_1, X_2,...,X_n$ is a random sample from $U(0,\theta)$. Find $E(X_{(n)}-X_{(1)})$. I attempted this question by first finding the CDF of $X_{(n)}-X_{(1)}$ using the formula: $$F_{U}(u)= n\int_0^\...
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If X~Exp(λ), what is the expected value of Y=X²?

I am trying to compute this using the integral definition of expected value but I don't think I am doing it right as I am getting a very hard integral that I can not solve. When computing $\mathbb{...
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Coordinate Ascent for Variational Inference: Deriving Updates

I am working with the following model and am attempting to derivate coordinate ascent updates using mean field variational inference: Sample $p_X \sim Beta(\alpha_1, \alpha_2)$ Sample $p_Y \sim Beta(...
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Expectation of precision, recall, f1

Let $X^n$ be a sample of size $n$ drawn from a Bernoulli distribution with mean $\rho$. Let $Y^n$ be a sample of predictions drawn from another Bernoulli distribution with mean $\gamma$. It's easy to ...
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Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\...
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How do the means of $X^2$ and $X$ compare?

If $X$ has an exponential distribution with mean $\theta$, does $X^2$ have mean $\theta^2$? If not, how would I find the variance of $X^2$? I tried this: $$V(X^2) = E[X^4] - E[X^2]^2$$ But I'm ...
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How do I find the expected value of F(isher)-distribution

$E(F)=\int xf_{k,m}dx$ where $f_{k,m}(t) = \Gamma(t)=\frac{\Gamma((k+m)/2)}{\Gamma (k/2)\Gamma(m/2)}k^{k/2}m^{m/2}t^{k/2 - 1}(m+kt)^{-(k+m)/2}$. How do you find $E(F)$? Say you have to convert $x*f(k,...