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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Combining two noisy measurements [duplicate]

Suppose I have two noisy measurements of a state (e.g. a radar detecting the position of an object). I model each measurement as a normal distribution, and I know the mean and variance of each ...
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The effect on expectation of biasing a distribution by a monotonic function

Let $$ x \sim g(x)$$ where $g(x)$ denotes the pdf of $x$. Let the pdf of another variable $x^*$ be denoted by $f(x^*)$ and let $$f(x^*) \propto g(x^*) z(x^*) $$ where $z(x^*)$ is a monotonic ...
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How do we determine the marginal mean using the joint PDF?

I'm reading chapter 4 of DeGroot and either I missed it or he didn't explain it but I am not seeing how, given a joint distribution f(X, Y), how do we find E[X] and E[Y]. I have learned how to find E[...
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Inverse moment of Multivariate Normal Norm

Let $x \sim N(\mu, \Sigma)$ with $\mu \in \mathbb{R}^P$; $\Sigma \in \mathbb{R}^{P\times P}$, positive definite. Denote by $||x||_2$ the vector two norm, that is, $||x||_2 = \sqrt{x^\top x}$. What ...
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Tractability of Expectations: log expectation vs expectation log

I'm working my way through a paper about bounds on the mutual information [1]. However, I have some issues in understanding claims they make about the tractability of the different bounds. Given: $ ...
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Intuition behind probabilities for sampling without replacement

I am not understanding the explanation here in regards to expectations and probabilities for sampling without replacement. I am not seeing why we don't treat each draw as a unique random variable. ...
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What is the mean and standard deviation of $5x^3 + 8$ when $X$ is the sum of numbers of two fair dice

Consider the following experiment: Throw two fair dice sequentially and define a random variable $X$ as the sum of the numbers of the two dice. Please compute the mean and standard deviation of $5X^3+...
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X follows N($\mu$, $\sigma^2$), then show that $E(x - \mu)^{(2k+1)} = 0$ [duplicate]

I tried using induction For $k=0$: $E(x-\mu) = E(x)-\mu = \mu-\mu=0$ assuming it holds for n= k so assuming $E((x-u)^{(2k+1)}) = 0$ we have to prove for $n= k+1$ $E((x-\mu)^{(2k+1+2))}$ = $E((x-\...
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Expected value of the impact of an event based on Poisson Distribution [closed]

I am using Poisson Distribution to calculate the following, Suppose we are measuring that a distribution company has on average 10 truck accidents per year, on average each accident costs 25.000$ ...
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33 views

How to represent skewness(X) in terms of the expected value?

Let $X$ be the random variable. $E(X)$ is the expected value of $X$ Then $Var(X)$ = $E(X^2)$ − $[E(X)]^2$ where $Var(X)$ is the variance of $X$ Then how to represent skewness(X) in terms of the ...
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Calculating expected value of gamma-distributed random variable [duplicate]

I am just looking for a quick check whether my reasoning is correct when calculating $E(X)$, given that $X \sim \Gamma(\alpha, \beta)$. My calculations are as follows: \begin{align*} \text{E}(X) &...
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Value of Regression Coefficient of Conditionally Independet Variable

Suppose we known that $E[Y\mid X=x]=f(\alpha + x\beta)$ for some pararmetic prediction method $f$. Consider now $E[Y\mid X=x, Z=z]=f(\alpha + x\beta + z\gamma)$. If $Y \perp\!\!\!\perp Z\mid X$ under ...
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Expected value of the F-Distribution dependent on the degrees of freedom associated with the Chi-square random variable in the denominator?

0 The F-Distribution has a Probability Density Function that can be defined as: with an expected value of: What would be a logical explanation for why the expected value only depends on the ...
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What is the expectation of the product of 2 random variables (Gauss-Markov assumptions)?

In the two variable (intercept and slope) model: among other, one of the Gauss-Markov assumptions is (in the BLUE framework of OLS) My coarse slides state this implies that there is no correlation ...
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Constructing random variable with specific expected value

Lets say I have a random variable $X$ and we don't know what the expected value is other than that it is some number $J∈[0,1]$. Lets say I'm interested in the expression $2J$. Without knowing $J$ I ...
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markov's inequality generalizability

Let $X : \Omega \rightarrow \mathbb{R}$ be a non-negative random variable on probability space $(\Omega, \mathscr{A}, P)$ and let $c > 0$. Then: $$\mathrm{P}[X > c] \leq \frac{\mathbb{E}(X)}{...
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Minimizing expected loss in Regression with Rademacher random variables

I am trying to prove the following equality. I am able to solve the terms inside the expectation but I am stuck because of the expectation with respect to $x,y$. I might be wrong in the whole process; ...
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If $cov(x_i,T_i)>0$ can I show $\mathbb{E}[\frac{T'x}{T'T}] > 0$?

x,T are vectors with $cov(x_i,T_i)>0$. Without specifying f(x,T), is it possible to determine the sign of $\mathbb{E}[\frac{T'x}{T'T}]$?
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Expectation and variance notation for ratio of random variables [duplicate]

When we have a ratio of random variables, is their expectation/variance defined in the same way? That is, if we want to write out explicitly $E[\frac{X}{Y}]$ where X and Y are random variables, then ...
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How to know which one is random variable?

Given a time series equation:$$x_t=\beta_1+\beta_2t+w_t$$ where $w_t$ is white noise. When I am calulateing the mean:$$E[x_t]=E[\beta_1+\beta_2t+w_t]$$ $$=\beta_1+\beta_2t+E[w_t]$$ $$=\beta_1+\...
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OLS as approximation for non-linear function

Assume a non-linear regression model \begin{align} \mathbb E[y \lvert x] &= m(x,\theta) \\ y &= m(x,\theta) + \varepsilon, \end{align} with $\varepsilon := y - m(x,\theta)$...
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MLE of variance is biased in a Gaussian distribution

Referring to: How to understand that MLE of variance is biased in a Gaussian distribution at some point during calculation the formula of the sum of the expected value becomes a single expected value:...
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Why does the expected value of the l=lag time series observation multiplied by the white noice equal zero in a stationary autoregressive model?

Suppose we have a stationary AR(1) model. Therefore $y_{t}=b_{0}+b_{1}y_{t-1}+e_{t}$ where e is the white noise. I am just wondering why (A) $E[(y_{t-l}-\mu)e_{t}]=0$. At first, I thought that $y_{t-l}...
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Expected HIGH and LOW of a random walk?

Given the random walk $s_n$ $$s_n=\sum x_i, \space\space\space x_i \text{ iid, }\space\space x_i \sim N[0,\sigma]$$ what are the expected highest/lowest values of the walk after $n$ steps? $$H_n=E[...
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37 views

Variance and expectation of $\frac{1}{n}\sum^n_{i=1}X^2_i$

Let $X = (X_1, . . . , X_n)$ consist of independent and identically Normal $N(0, θ)$ random variables, with mean $0$ and variance $θ \gt 0$. The Moment Estimator for $\theta$ is given by $\hat \theta ...
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1answer
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Find expected value with pdf and LOTUS

I am currently trying to solve a problem and can't figure it out. I have done this before, but I can't remember all of the details and can't find a reference example. Let's say I have a pdf $$f(x)=\...
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Expectation and variance of the squared distance between $X$ and $Y$

Given that $X$ and $Y$ are two independent univariate random variables sampled uniformly from the unit interval [0,1]. I am trying to find the expected value and the variance of the random variable $Z ...
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How is that possible that simple arithmetic mean works well even for strongly skewed distribution?

I was taught, that the arithmetic mean is sensitive to outliers and skewness. This was natural to me - the observations lying far from the "central point" of the distribution "pull" the measure ...
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Forecasting with probabilities

I need help with how I should approach a forecasting problem. I have two tables like the ones below. ...
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23 views

paradoxical manipulation based on law of total probability

I am trying to resolve the following paradoxical result when I am using law of total probability to manipulate some expression, but can't seem to be able to. Suppose we have random variable $X, Y, Z$...
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Average time ant needs to get out to the woods

An ant has three passages to choose from: Passage A takes 7 minutes to get ant out of the ant house to the woods. Passage B takes 8 minutes to get ant back to the starting point where he is. Passage ...
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Endogeneity and law of iterated expectation

My understanding of the endogeneity issue is that $\mathbb{E}[Y|X]$ is inconsistently estimated. For instance if we know $\mathbb{E}[Y|X] = \theta X$, then the estimator $\hat{\theta}$ is inconsistent ...
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Expected value $\mathbb{E}[X-Y \mid X>Y, Y< c]$ for independent variables $X,Y$?

I am looking for the expected value $\mathbb{E}[X-Y \mid X>Y, Y< c]$ for independent variables $X \sim U(x_1, x_2),Y \sim U(y_1,y_2)$, $c\in [y_1, y_2]$? Is the following formula correct? $$\...
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1answer
23 views

elementary coin-toss problem

Toss a coin 10 times. Find the expected number of heads in the first 5 tosses, given 6 heads in the 10 tosses The given solution: Think of a box containing 6 heads and 4 tails. Draw 5. The ...
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Prove that expected test error is the same regardless of number of training points in the dataset

This seems to make sense, since the expected value for predicted points would always be the same constant regardless of the number of training points, but how can I prove this mathematically?
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Is $E[g(X_1,…,X_n)] = E[E(g(X_1,…,X_n) | X_2,…X_n)]$?

I know that $E(X_1) = E[E(X_1 | X_2)]$, but I’m wondering if I can generalize this to $E[g(X_1,…,X_n)] = E[E(g(X_1,…,X_n) | X_2,…X_n)]$ based on the following: $$E(g(X_1,…,X_n)) = \int_{-\infty}^{\...
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Expected of number of discrete uniform variables whose sum is bigger than k (from characteristic function of discrete Irwin–Hall distribution?)

The problem Imagine we keep uniformly drawing $n$ integers $X_i$ from {0, 1, ..., 9} so that their sum is more than $10$. For instance, one draw would be {1, 0, 2, 5, 3}, hence $n=5$, and repeat this ...
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Is it correct to write $\Bbb E[X]$ or $\Bbb E_{\theta}[X]$?

Suppose that we observe the discrete random variable $ X = (X_1, \dotsc , X_n)$ with state space S, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known ...
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1answer
28 views

How to find the MGF of the difference of 2 random variables

Let X ~ N(12,4) and Y ~ N(3,1) Let Z = X - Y Find the Moment Generating Function of Z. I tried finding the expected value of e to the power of tz, but this isn't possible to separate in the ...
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Bayes - Integral to Gamma Function

When trying to understand my professors notes, I came upon this piece and don’t understand the step he took. $\frac{(\sum x_i+2)^{n+1}}{\Gamma (n+1)} \int_0^\infty\theta^{n+1} e^{-\theta \sum x_i+2}\...
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1answer
60 views

Expected value with dependent samples

It is well known that the expected value of a function can be approximated with i.i.d. samples: $$ E_X[f(X)] = \frac{1}{n}\sum_{i=1}^n f(x_i),\quad x_i\sim_{i.i.d.} X $$ What methods exist to ...
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How to approximate expectation and variance of an integral from a discrete Time series financial dataset?

I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this. ...
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1answer
40 views

How to solve for conditional expectations with binary random variables?

Suppose you have a binary random variable X with a probability of 50% 1, and a probability of 50% 0. Another random variable Y has the conditional expectation E[Y|X]=5. Another random variable W=XY-X. ...
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1answer
53 views

Mean and variance of a Random Walk with lower boundary

Consider a random walk $S_n$ that starts at $S_0$ and terminates if it hits a lower boundary of $0$. $$ S_n=\begin{cases} 0, & \text{if } S_{n-1}=0\\ 0, & \text{if } S_{n-1}+x_n\le0\\ S_{n-...
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1answer
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Likes/Retweets in Twitter data depends on time of extraction

Currently I scheduled a task to extract twitter data of a week (every sunday) to predict the stock market for the following days. The number of likes of a tweet is not static and changes over time. ...
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2answers
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Approximating the expected value of a random variable as a function of the prior mean of a parameter

I have a parameterised statistical model and I'm trying to calculate the expected value of a random variable. I know that the expected value is a function of the value of the unknown parameter. So I ...
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What is the distribution of sum of Gamma and Chi square distributed dependent random variables?

Let $\mathbf x=[x_1, ... ,x_K]^T$, $x\sim\mathcal C\mathcal N(\mathbf 0,\sigma_x^2\mathbf I)$, and $\mathbf y=[y_1, ... ,y_K]^T$, $y\sim\mathcal C\mathcal N(\mathbf 0,\sigma_y^2\mathbf I)$. Consider $...
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1answer
25 views

How to analyse a contingency table with specific expected distribution?

I have a 2x2 table of single nucleotide substitutions in DNA. The data rows represent the nucleotides in one genome (A) and columns are the nucleotides in the same positions of a different genome (B). ...
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21 views

What is the distribution of squared Erlang random variable?

Let $\mathbf x=[x_1, ... ,x_K]^T$, $x\sim\mathcal C\mathcal N(\mathbf 0,\sigma_x^2\mathbf I)$, I believe that the distribution of $||\mathbf x||^2=\mathbf x^{\dagger}\mathbf x$ is Erlang. Is there ...
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1answer
38 views

How to express the expectation of a vector of Bernoulli random variables?

I am reading this paper https://arxiv.org/pdf/1703.07370.pdf The problem setup is as follows: I am trying to express the expectation, $$\mathbb{E}_{p(b)}[f(b,\theta)]$$ into a summation. For ...

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