# 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, $\displaystyle X_{RMS}=\sqrt{{1 \over N}(x^2+x^2+...+x[N]^2)}$, 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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### 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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### 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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### 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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### 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 ...
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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 $$\mathbb{E}(Z|Z>\tau) >... • 173 4 votes 1 answer 36 views ### Finding most probable vector, given angles Assuming you have a series of known vectors k_i in \mathbb{R}^N, each at an angle \theta_i to a single unknown vector v. All vectors have the same dimension N which can be arbitrarily ... • 73 0 votes 1 answer 42 views ### Does conditional expectation agree with joint expectation? Suppose I have a vector valued random variable (X_1, X_2) in \mathbb{R}^2, with density \pi(x_1, x_2). Let \mathbf{\mu} = (\mu_{X_1}, \mu_{X_2}) denote the mean vector of this bivariate ... 3 votes 1 answer 104 views ### Calculating expected value from quantiles For probabilities p_i=\frac{i}{10} where i=1, \dots, 10, the respective quantiles are \tau_i. How can I calculate an approximate expected value? • 224 2 votes 0 answers 34 views ### How to (dis)prove \lim_{k\to\infty}\lim_{n\to\infty}E(Y_{n,K}) = E(\min(X_n, K))? Here is the problem: Given X, X_1, X_2, \ldots, non-negative random variable with finite expectation and X_n \to X pointwise and Y_{n,K} = \min(X_n,K), we are asked to see if a) \lim_{K \to \... • 408 4 votes 2 answers 148 views ### Expected squared dot product between IID Gaussian vectors? Suppose x,y are IID samples from a Gaussian distribution in \mathbb{R}^d. The following seems true:$$2\ \mathbb{E}\left[\langle x, y\rangle^2\right] = \mathbb{E}\left[\|x\|^4\right]-\mathbb{E}\...
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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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### 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 $E[X^2YZ] = E[X^2]E[Y]E[Z]$