Questions tagged [expected-value]

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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$E[X^T (Y-Z)] = E[X^T] E[Y-Z]$ but what about $E[(X^T (Y-Z))^2]$?

Let $X, Y$, and $Z$ be random vectors with $X$ independent of $Y$ and $Z$. Due to the independence we have $$ E[X^T (Y-Z)] = E[X^T] E[Y-Z]. $$ But what what $E[(X^T (Y-Z))^2]$? Is it possible to ...
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Expected value of a conditional normal [closed]

Why $E[(Y- E[Y|Z])^2] = \sigma^2_Y(1-\rho^2)$????? I know this is the variance of the conditional distribution $Y|Z$. I thought $E[(Y|Z- E[Y|Z])^2] = \sigma^2_Y(1-\rho^2)$, but the slides says ...
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Expected value of stochastic process given probability of number of sign changes on interval

We have a stochastic process $X_t$, which at a given time $t$ have a value of $-1$ or $1$. Number of sign changes on an interval $(t; t + \Delta)$ have a Poisson distribution $P(N = k) = e^{-\lambda\...
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Finding correlation coefficient of $X$ and $XY$

Let $X$ and $Y$ be independent random variables with nonzero variances. I'm looking to find the correlation coefficient $\rho$ of $Z=XY$ and $X$ in terms of the means and variances of $X$ and $Y$, i.e....
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If $Y|X \sim N(0,1)$, does that mean $Y$ itself is also $N(0,1)$? [duplicate]

If $Y|X \sim N(0,1)$, does that mean $Y$ itself is also $N(0,1)$? It looks like it as I can write $$ E[Y] = E[E[Y|X]] = E[0] = 0, $$ and $$ Var[Y] = Var[E[Y|X]] + E[Var[Y|X] = Var[0] + E[1] = 1. $$ ...
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The expectation of the inverse of a negative binomial random variable?

Suppose $X \tilde{} NB(n,p)$ and $\mathbb{P}(X=x) = \binom{x-1}{n-1}p^n(1-p)^{x-n}$. Then what is $\mathbb{E}(\frac{1}{X}) = \sum_{x=n}^\infty \frac{1}{x}\binom{x-1}{n-1}p^n(1-p)^{x-n}$? Many thanks
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Finding expected value of operating units [closed]

Given $N \cdot k$ operating units, every unit consist $k$ elements. If one element is missing from one operating unit, it stops working. Choosing $m$ element randomly, what will be the expected value ...
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How to calculate the expected value of a time series just from the data

in this question Can stationary time series contain regulary cycles and periods with different fluctuations I was told that stationary time series do not have regular cycles and that having constant ...
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Boundary of $E\left[\frac{\prod_{i=1}^n x_i}{\prod_{i=1}^n x_i+\prod_{i=n+1}^m x_i}\right]$

Suppose $X_i$ are i.i.d. In addition, $X_i>0$ and $E[X_i]>1$. Suppose $E[X_i]$ is known, could we find upper bound or lower bound for the following expectation: $$ E\left[\frac{\prod_{i=1}^n x_i}...
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Finding variance of the quotient of normal distribution and chi-squared distribution

Given that $Z\sim N(0,1), Y \sim \chi^2_{v}$, and assuming that $Z, Y$ are independent, we define $W=\frac{Z}{\sqrt{Y}}$. I aim to find $E(W)$ and $Var(W)$, with possible defining of $v$. Finding $E(W)...
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The for Marginal pdf and Expected Value for 2 Random Variables with Joint Probability Distribution

I have trouble with determining the domain for integration in the case of having a joint pdf when one variable depends on the other. There are two examples I don't quite understand, and hopefully, you ...
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Conditional expectation of two random variables

Suppose we have $X \sim Exp(4)$ and $Y \sim N(0,1)$ which are independent. What can we say about $\mathbb{E}[X|Y]$?
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If $Y|X \sim \mathcal{N}(0,1)$ then is $Y^2|X \sim \chi^2(1)$?

Suppose we have random variables $X$ and $Y$ such that $Y|X \sim \mathcal{N}(0,1)$. Can we then say that $Y^2|X \sim \chi^2(1)$? If we can, then what about when $Y|X \sim \mathcal{N}(0,\sigma^2/4)$, ...
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Computationally estimate $E[f(\hat \beta_1 X)]$ where $\hat \beta_1$ is the estimated coefficient obtained by ordinary least squares regression?

Let $(X_1,Y_1),(X_2,Y_2),\dots,(X_5,Y_5)$ be i.i.d samples and consider the regression model $$ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i, \quad \quad \text{for} \ i \in \{1,2,\dots,5\}, $$ where $\...
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can you guide me how can ı solve this?

Missiles are launched until one successfully reaches the target. If the expected number of launches is 2.5, find the probability that at most 3 attempts will be needed.
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Find some expectation of two random variable given the joint distribution

I have the following joint distribution for the random variable (X,Y): Now, I have to compute the following expectations: $E(X|Y)$ $E(X+Y^2 | Y)$ Now, for the first point, the procedure was the ...
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Calculating the conditional expectation of an exponential family [closed]

If we have $X$ with a density depending on the scalar parameter $\theta$, where the density is from of the exponential family: $f(x;\theta) = \exp(\theta x−\phi(\theta))h(x)$. Also we have that $\...
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How can I best calculate the expected amount of life lost over the next 50 years due to a constant yearly 1.5% risk of death?

Say that Person X has 50 years left to live until they die from cause Z. Cause Y is a constant 1.5% risk of death per year for the next 50 years. I want to determine what the expected amount of life ...
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Monte Carlo integration with Control Variate not giving any variance reduction

Given $I=\int_{3}^{+\infty}\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2} \, dx$, I want to use a Control variate function to reduce the variance of a simple Monte Carlo simulation to compute this integral. In ...
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If a random variable $Y$ converges in distribution, can we use the parameters of the asymptotic distribution as if they are the parameters of $Y$?

Let $Y_n$ be a sequence of random variable such that $$ \sqrt{n}(Y_n-\mu) \stackrel{d}{\to} \mathcal{N}(0, \sigma^2), $$ and thus we can say $Y_n$ is asymptotically normally distributed as $$ Y_n \...
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Number of heads in 100 Coin toss [closed]

I got a coin that follows a certain distribution (with the probability of head). I toss the coins 100 times. What is the correlation between: number of heads I got (in the 100 toss). number of heads ...
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E[X/Y], iid, show it is greater than 1 [closed]

X, Y are IID. show E[X/Y] > 1.
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Trying to approximate $E[f(X)]$ - Woflram Alpha gives $E[f(X)] \approx \frac{1}{\sqrt{3}}$ but I get $E[f(X)] \approx 0$?

Let $X \sim \mathcal{N}(\mu_X,\sigma_X^2) = \mathcal{N}(0,1)$. Let $f(x) = e^{-x^2}$. I want to approximate $E[f(X)]$. Wolfram Alpha gives \begin{align} E[f(X)] \approx \frac{1}{\sqrt{3}}. \end{align} ...
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Why is the observed Fisher information defined as the Hessian of the log-likelihood?

In an MLE setting with probability density function $f(X, \theta)$, the (expected) Fisher information is usually defined as the covariance matrix of the fisher score, i.e. $$ I(\theta) = E_\theta \...
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I have distribution that range from 0-1.8 and mean E[X] I want to move the mean by y How I modify original distribution so that it has mean E[X] -y?

In a way, I want to change the original distribution so that it has desired mean of E[X]-y without shifting the distribution by y amount. Is there a way to do this?
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Minimizing $L^p$ distance of a random variable and a constant

Let $X$ be a real-valued random variable. Then expected value $\mathbb{E}[X]$ is the number $c$ minimizing $c \mapsto \mathbb{E}[(X-c)^2]$. Similarly, the median of $X$ is the number $c$ minimizing $c ...
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If $\hat \alpha \sim N(\alpha,\sigma)$, is $E[\phi(\hat \alpha X)] = E[\phi(\alpha X)]$ where $\phi$ is the Gaussian function?

Let $\hat \alpha \sim N(\alpha,\sigma)$. Obviously $$ E[\hat \alpha - \alpha] = E[\hat \alpha] - E[\alpha] = \alpha - \alpha = 0. $$ Now define $\phi$ to be the Gaussian function $$ \phi(x) = e^{-x^2/...
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Given expectations and variances of random variables check if they can be independent?

I have the following exercise: Knowing that $E[X] = E[Y] = E[Z] = 0$ and $E[X^2] = E[Y^2] = E[Z^2] = 1$. Random variables $X, Y - X, Z - Y$ are independent. Can $X, Z$ be independent? Can someone ...
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In real-life applications, which continuous distributions have NON-CONVERGENT expectations that require Lebesgue integration?

When computing expected values, Riemann integration works for only random variables with bounded support sets. For distributions with unbounded support sets, we can use improper Riemann integrals for &...
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Trying to show $E[\hat \beta_1 | \mathbf{X}] = \beta_1$ directly from the definition of $\hat \beta_1$?

Suppose we have the standard simple linear regression model: $$ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i, $$ with $E[\varepsilon_i|X_i] = 0$ and $\text{Var}[\varepsilon_i|X_i] = \sigma^2$. I'm ...
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Is $E[Y (X_1+X_2)^2| \mathbf{X}] = (e^{3X_1}+3X_2) E[Y | \mathbf{X}]$ for $\mathbf{X} = [X_1 X_2]^T$?

Let $\mathbf{X} = [X_1 X_2]^T$ be a vector of random variables and let $Y$ be another random variable that is dependent in some way on $X_1$ and $X_2$. Suppose we want to calculate $E[Y (e^{3X_1}+3X_2)...
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Expected value of a bivariate distribution as an integral

Let's assume an absolutely continuous random variable, $X$, with PDF $f(x)$. $$\mathbb{E}\big[X\big] = \int_{\mathbb{R}}xf(x)dx$$ If $X\sim f(x_1,x_2)$ is multivariate, then it makes sense to me to ...
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Obtaining expectation of a variable given a conditional expectation

Is there any mathematical association between the conditional expectation of a variable given another variable, and the unconditional expectation of that variable? I realise that given a joint ...
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Expected value (Mean) of a joint distribution

I saw in a textbook that if we have a joint distribution $f(X,Y)$ that is a Gaussian distribution, then we have the mode equal to the mean. The mode is just the values of $X$ and $Y$ such that $f(X,Y)$...
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Consistency of likelihood importance sampling estimator

In a lecture recently our lecturer described a method for approximating the expectation of a function over a posterior distribution using likelihood importance sampling. That is: $$ \mathbb{E}_{p(x|D)}...
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Expected value and standard deviation of roulette

Suppose you are playing multiple independent rounds of roulette where you bet $x$ dollars on black then with probability $18/38$ you will double that bet (net gain of $x$ dollars) and with probability ...
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Estimate limit on how much bigger one group is than another

I have two groups of independent, unpaired data. I would like to estimate how much bigger one group is than the other on average - e.g. being able to say that group 1 is no more than 50% higher than ...
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Memoryless conditional expectation of shifted function exponential

Related to this, is the following valid: \begin{align} E[f(X-t) \mid X>t] = \int f(y-t) f_{X|X>t}(y) dy = \int f(x) f_{X|X>t}(x+t)dx = \int f(x) f_X(x) dx = E[f(X)] \end{align} where I make ...
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upper bound of mean absolute difference

Let $X$ be an integrable random variable with CDF $F$ and inverse CDF $F^*$. $Y$ is iid with $X$. Prove $$E|X-Y| \leq \frac{2}{\sqrt{3}}\sigma,$$ where $\sigma=\sqrt{Var(X)}$. I am looking for some ...
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Expectation of the product of polynomial & exponential transformations of normal r.v

Let $X \sim \mathcal{N}(\mu, \sigma^2)$. Are there any (1) general formula and (2) references to the general formula for $$ \mathbb{E} (X^n e^{tX}),\; n \in \mathbb{N}, t \in \mathbb{R}$$ in ...
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Expectation with respect to a transformed random variable

Problem Suppose I have a random variable $z$ following a distribution $p(z)$. Suppose I have a transformation $$ f(z) = x $$ that transforms the random variable $z$ into a new random variable $x$ with ...
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covariance of squared projections

Given a vector $x$ of independent mean-zero random variables, and two nonrandom orthogonal unit vectors $u,v$, does $u'v=0$ imply $cov(x'uu'x,x'vv'x)=0$? If so, what is the proof? If not, what happens ...
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Symmetry of distribution function is defined as $f(x-a)=f(-(x-a))$, then expectation is $'a'$. i.e $E(X)=a$ [duplicate]

I came across this statement in a book. While I know, how to prove $E(X) = a$ is using $f(x+a)=f(x-a)$. I cannot seem to prove it using $f(x-a)=f(-(x-a))$. I keep going on in a loop, no matter what I ...
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How many rolls required for 90% chance to reach expected values of consecutive dice rolls in tabletop game?

I'm trying to work out if random variance in dice rolls is more likely to influence a given situation in a game rather than the overall expected values of those dice rolls being significant. The game ...
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Expectation of the exponential function of Absolute Value of the Difference of Two Double parameter exponentially Distributed Random Variable

Suppose, $X_1,\ldots, X_n$ be iid having double parameter Exponential Distribution with common pdf $$f(x)= \dfrac{1}{\sigma} \exp\{ -(x-\mu)/\sigma \} I(x>\mu); \mu \in R, \sigma \in R^+ , n\ge5$$ ...
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1answer
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Covariance of Sum of Random Variables [closed]

Let $a_1,a_2,b,c_1,c_2,d$ be constants and assume $X_1, X_2, Y_1, Y_2$ are Random Variables. I am trying to prove $$Cov(a_1X_1+a_2X_2+b, c_1Y_1+c_2Y_2+d)= a_1c_1Cov(X_1,Y_1)+a_1c_2Cov(X_1,Y_2)+...
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Problem calculating expectation using law of total expectation

I'm confusing myself with conditional expectation and could really use your help! I am trying to calculate an expectation that arises in the context of doing variational inference. However, the ...
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1answer
29 views

Independence of variables between X and Y^X

If X and Y are independent, then are X and Y^X independent? Does the realisation of X have to be the same as the X in the power of Y? I think this question sounds silly but I'm trying to clear a major ...
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Evaluating (Uniform) Expectations over Non-simple Region

Background. Let $V = (X,Y)$ be a random vector in 2-dimensions uniformly distributed over two disjoint regions $R_X \cup R_Y$ defined as follows: $$ \begin{align} R_X &= ([0,1] \times [0,1]) \...
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1answer
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Modelling Roulette

I'm trying to model a series of plays on a game of American roulette. This is where you can't bet on two numbers (the zeros) rather than European Roulette where you can't bet on one. If you bet on ...

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