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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Two fair dices are rolled, say X_1 and X_2, respectively. Let Y = X_1 + X_2. What is the expected value of Y, given that X_1 and X_2 is not equal? [closed]

Need help preparing for my exam, thank you! I also worked out that the E[Y] = 7, and E[Y | X_1 and X_2 are even] = 8, are these answers correct?
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Discrepency between trials and successes for varying probability of success

Suppose there are $N$ balls in an urn, with $X$ white balls and $N-X$ black balls. We perform $k$ iterations of the following process: Choose a random ball form the urn. If the ball is white, we put ...
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What is the expected value of the logarithm of a variable of the Generalized Dirichlet?

For a Dirichlet $\operatorname{Dir}\left(\mu_1, \cdots, \mu_K \mid \alpha_1, \cdots, \alpha_k\right)$, \begin{equation} \mathbb{E}\left[\ln \left[\mu_j\right]\right]=\psi\left(\alpha_j\right)-\psi\...
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Given that X and Y are normally distributed as N(0,3) and N(0,5) respectively, what is the expected value of (XY)^2?

Given that X and Y are independent and normally distributed as N(0,3) and N(0,5) respectively, what is the expected value of (XY)^2?
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How do we relate RMS and standard deviation for continuous signals?

Because the discrete formula for RMS, $\displaystyle X_{RMS}=\sqrt{{1 \over N}(x[1]^2+x[2]^2+...+x[N]^2)}$, is almost the same as the formula for standard deviation (assuming mean zero), except for a ...
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In statistics how does one find the mean of a function w.r.t the uniform probability measure?

I am unfamiliar in statistics. My knowledge is in pure mathematics. Suppose $n\in\mathbb{N}$, where $X$ is in the $\sigma$-algebra of Caratheodory-measurable sets such that $X\subseteq\mathbb{R}^{n}$ ...
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Expected value of the max of scaled powers of the same random variable

I've been looking at order statistics and the behavior of expectations when max is involved, but that literature always discussed iid random variables whereas I have a strange situation where I have ...
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What is the probability the expected value is undefined or infinite?

What is the probability from a uniform probability measure (pg.37) on sample space $\left\{N(\theta,1)|\theta\in[0,1]\right\}$ that for some random variable $X$ in the sample space, the Expected-Value ...
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2 answers
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Possible typo in discussion of moments of a random variable

I'm struggling to understand some notation in this excerpt from Larsen & Marx. Under "Comment" j is defined as ...
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Lower bound on gradient of the restricted function

Let $A$ be an $N\times n$ matrix where $N < n$ and $b$ be a vector in $\mathbb{R}^N$. Let $f: \mathbb{R}^n \to \mathbb{R}$ be a quadratic function $f(x)=\frac{1}{N}\|Ax-b\|^2$. Define two sets $J \...
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$X\sim N(0,1)$. $E(X^2)=?$ have two different results for me. one is zero other is 1

I have a confusing result: independent $X\sim N(0,1)$ and I'm looking for $E(X^2)$: $E(X^2) = E(X \times X)$ $E(X^2) = E(X)\times E(X)$ $E(X^2) = 0 \times 0$ $E(X^2) = 0$ I can also do this: $Var(X) = ...
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Expected value of applying the product of the sigmoid-like function to a normal distribution

Let $\sigma : \mathbb{R}\to\mathbb{R}$ be a sigmoid-like function, $b \sim \mathcal{N}(0, 1)$, $x_1 \in \mathbb{R} $, and $x_2 \in \mathbb{R}$. Can we obtain a closed-form solution of $\mathbb{E}[\...
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Notation confusion regarding Expectation in Kullback-Leibler divergence definition

KL-divergence is often defined as $$D_{KL}(P||Q) = E_{x\sim P} \left[ \log\left(\frac{P(X)}{Q(X)}\right) \right] = \int_{-\infty}^{\infty} P(x) \log\left(\frac{P(X)}{Q(X)}\right)dx$$ I don't quite ...
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Compute Expectation and Variance of number of upper records

I should solve the following: Let $X_i,...,X_n$ be independent random variables such that for any i, j, i$\neq$j: $P(X_i=X_j) = 0$. Then, $X_i$ is an upper record iff $X_i > \max$ {$X_1,X_2,...,X_{...
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what is the expecxted length of the initial run for heads? [duplicate]

Suppose a coin is tossed repeatedly with a probability of head appears in any toss being $p,~0<p<1$. I want to find the expected length of the initial run of heads. Here initial run for heads ...
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8 votes
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When does the expected value or variance of the $t$ statistic exist?

The distribution of Student's $t$ statistic is known when the random variable $x$ follows a Normal distribution. Sometimes, however, we apply it to random variables drawn from other distributions. I ...
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Estimating the integral $\int_{0}^\infty x^4 e^{-2x}\,dx$

We have been given a random variable having a Gamma distribution as shown below: Using the accept-reject algorithm, we are supposed to sample from the Gamma distribution using exponential as the ...
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Probability of net gain being greater than 0 lottery question

I'm working on some practice exams and I came across a probability question that stumped me. "In a lottery game, for any lottery ticket bought at random, the chance of winning each prize is ...
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Equivalent of $E[(a-X)^2] = E[(a-E(X))^2] + Var(X)$ for $E[|a-X|]$ and $med(X)$?

The minimzer of the MSE $E[(a-X)^2]$ is $a=E(X)$, and the MSE can be decomposed into $E[(a-X)^2] = E[(a-E(X))^2] + Var(X)$. I am wondering whether there exists a similar expression th MAE $E[|a-X|]$ ...
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Questions about the expected target value $\bar{y}(x)$

Assume a simple Linear Regression problem, where we have $n$ data points $x$, and one target variable $y$. My confusion, or more precisely misconceptions start at the following equation and its ...
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$ E(U_x) $ when $ U_x $ is the number of smaller values in $ X , Y $

> I have added my solution here $ U_x $ be the number of $ y's $ those are smaller than $ x's $ in independent random samples $ X_1, X_2 ,\ldots, X_n $ and $ Y_1, Y_2,\ldots,Y_m ;$ find $ E(U_x)$...
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Is there a name for the "expectation" w.r.t. the survival function?

I am doing survival analysis and was wondering if there was a name for the following expression: $$H(t)=\int_{0}^{\infty}{g(t) \cdot S(t) \cdot dt}.$$ It appears to be very similar to the equation for ...
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Expectation as a minimizer of the loss function

It is a well-known fact that the minimizer of the mean-squared loss (MSE) $$\min\limits_\mu \mathbb{E}_{X} \left(X - \mu \right)^2$$ equals the expectation of $X$. Are there any alternative non-...
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calculating 'expected' rainfall for a period (e.g., month) when there is zero-inflation and over dispersion

I want to use 20 years of estimated precipitation data (maybe from CHELSA or CHIRPS datasets) to look at what the expected amount of precipitation is for different 30-day periods. The main purpose of ...
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2 answers
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Mean of the log and variance of the log

I am struggling to derive the following identities: $$ \mathbb{E}[\log Z]=2\log(\mathbb{E}[Z])-\frac12\log(\mathbb{E}[Z^2]) $$ $$ \mathrm{Var}[\log Z]=\log(\mathbb{E}[Z^2])-2\log(\mathbb{E}[Z]) $$ ...
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Expected value of inverse distance between two 3D normal distributions

Consider two independent trivariate normal random variables $X$ and $Y$. The means are non-zero and the off-diagonal elements of the covariance matrices are non-zero. $X$ and $Y$ does not follow the ...
2 votes
1 answer
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For random variable $Z=\max_i X_i$, can we bound $\mathbb{E}(Z|Z>\tau)$ with $\mathbb{E}(Z)$

Let $X_1,…,X_n$ be independent, but not necessarily identical, non-negative random variables. Let $Z=\max_i(X_i)$. Fix a real $\tau > 0$. Is there a way to lower bound $$\mathbb{E}(Z|Z>\tau) >...
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Finding most probable vector, given angles

Assuming you have a series of known vectors $k_i$ in $\mathbb{R}^N$, each at an angle $\theta_i$ to a single unknown vector $v$. All vectors have the same dimension $N$ which can be arbitrarily ...
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Does conditional expectation agree with joint expectation?

Suppose I have a vector valued random variable $(X_1, X_2)$ in $\mathbb{R}^2$, with density $\pi(x_1, x_2)$. Let $\mathbf{\mu} = (\mu_{X_1}, \mu_{X_2})$ denote the mean vector of this bivariate ...
3 votes
1 answer
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Calculating expected value from quantiles

For probabilities $p_i=\frac{i}{10}$ where $i=1, \dots, 10$, the respective quantiles are $\tau_i$. How can I calculate an approximate expected value?
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How to (dis)prove $\lim_{k\to\infty}\lim_{n\to\infty}E(Y_{n,K}) = E(\min(X_n, K))$?

Here is the problem: Given $X, X_1, X_2, \ldots$, non-negative random variable with finite expectation and $X_n \to X$ pointwise and $Y_{n,K} = \min(X_n,K)$, we are asked to see if a) $\lim_{K \to \...
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Expected squared dot product between IID Gaussian vectors?

Suppose $x,y$ are IID samples from a Gaussian distribution in $\mathbb{R}^d$. The following seems true: $$2\ \mathbb{E}\left[\langle x, y\rangle^2\right] = \mathbb{E}\left[\|x\|^4\right]-\mathbb{E}\...
2 votes
1 answer
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Inconsistency of Bayesian Premium

Given: The amount of a claim, $X$ is uniformly distributed on the interval $[0,\theta]$ The prior density of $\theta$ is $\pi(\theta) = \frac{500}{\theta^2}, \theta > 500$ Two claims, $x_1=400$ ...
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Expectation of ratio of 2 Normal(0,1) with one shifted [duplicate]

I have the following random variables $X$ ann $Y$ following respectively $N(0,b)$ and $N(0,c)$ $Z=\dfrac{(X+a)}{Y}$ with $a$ a real number. What's the expectation of Z, i.e $E(Z)$ ? UPDATE : sorry for ...
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Roll 4 Dice, What's the Expected Value of the Sum of the highest 3?

After writing a simulation in python (code at bottom) I realized my calculations are incorrect but can't figure out where I went wrong. Let {$D_1$, $D_2$, $D_3$, $D_4$} be the ordered dice rolls. Let $...
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Expectation of the product of two correlated normal variables [closed]

Let two independent normal variables $X\sim N(\mu_1, \sigma_1^2)$ and $Y\sim N(\mu_2, \sigma_2^2).$ Let $Z=e^X+e^Y.$ Then $Z$ is lognormal random variable. Is it possible to calculate $\mathbf E[X\...
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What is $E\left[\frac{X^2}{X^2+Y^2}\right]$ if $X$ and $Y$ are normally distributed but not iid.?

I assume that X and Y are normally distributed with individual mean and variance. So far, I have found that an analytic expression exists for $E[X^2+Y^2]$, $E[X^2*Y^2]$ and $E[X^2*(X^2+Y^2)]$, all ...
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Finding users who will buy. Intractable Numbers

I'm facing the following problem. I think I have it formulated right, but unsure how to proceed. Here is the problem statement. Assume there are $N$ users. For each user, I've an estimated probability ...
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A problem about Multivariate Normal Distribution

$(X,Y,Z)$ is a multivariate normal distribution. Calculate $E[X^2YZ]$ I'm finding an approach for this problem. I'm not sure if it is possible to assume $E[X^2YZ] = E[X^2]E[Y]E[Z]$
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How to calculate expectation of a discrete random variable using CDF? [duplicate]

Please refere to the attached image (source: Wikipedia). They have derived the formula (summation one) for discrete random variable as a special case. Can anybody please help me understand how to ...
2 votes
1 answer
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Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian

Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$. I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
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Show that $\min_{a \in \mathbb{R}} E \left[ \max \left( (1-a) V, a Z \right) \right]$ is minimized by $a$ such that $0<a<1$

Consider two independent random variables $V$ and $Z$. Assume that $Z$ is standard normal. We only assume that $E[|V|]<\infty$ and $E[V]=0$. Now consider the following optimization problem: \begin{...
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Mean (or lower bound) of Gaussian random variable conditional on sum, $E(X^2| k \geq|X+Y|)$

Suppose I have two mean zero, independent Gaussian random variables $X \sim \mathcal{N}(0,\sigma_1^2)$ and $Y \sim \mathcal{N}(0,\sigma_2^2)$. Can I say something about the conditional expectation $E(...
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1 answer
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How to find the expected value and median of a chi square distribution with $12$ dof in R?

How can I find the expected value and median of $X\sim \chi^2$ with degrees of freedom of $12\,$? The information I get from every other source, I find, is confusing. I am using R.
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Variance of positive random normal variable

Consider $R=X\mathcal{I}(X>0)$, where $\mathcal{I}$ is the indicator function and $x \sim N(\mu,\sigma).$ The mean $\mu_R = \mu \Phi(\frac{\mu}{\sigma}) + \sigma\phi(\frac{\mu}{\sigma}) $ is found ...
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2 votes
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Expectation of a positive random normal variable

Let $R=X\mathcal{I}(X>0)$, where $\mathcal{I}$ is the indicator function and $x \sim N(\mu,\sigma).$ Thus, \begin{align} \mathbb{E}[R]&=\mathbb{E}[X\mathcal{I}(X>0)] = \int_{-\infty}^\infty ...
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2 answers
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Expected value of random variable that is defined by another

Suppose I have a random variable: \begin{equation} X \sim \begin{cases} 1 \text{ with probability } p \\ y \sim \mathit{unif}(1,k) \text{ with probability } 1 - p \end{cases} \end{equation} Where $\...
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Can $e^{E(\log(x))}$ be calculated approximating $E(\log x) $ using a second order Taylor expansion around the mean?

While searching around, I found this question Expected value of a natural logarithm dealing with the expected value of the natural log. The top answer references a paper that approximates the expected ...
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Variance of the product of two normal variables [duplicate]

out of curiosity, I wonder if there is a solution for the expectation of the product of two chi-square variables (or the variance of the product of the normals). Say: Let $(X,Y)$ be jointly normal, ...
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3 votes
2 answers
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Show that $\Pr(G=1|Z=1, \eta)=1$ and $\mathbb{E}(\eta|Z=1)=0$ imply $\mathbb{E}(G\times \eta|Z=1)=0$

Take two binary random variables $G,Z$ and a continuous random variable $\eta$. Assume $$ \Pr(G=1|Z=1, \eta)=1 \text{ almost surely} $$ and $$ \mathbb{E}(\eta|Z=1)=0 $$ Could you help me to show that ...
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