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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32 views

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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17 views

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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15 views

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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Determine Expected Value and Bias of Gamma Distribution [closed]

Let $X_1,…, X_n$ be a sample of $iid$ $Gamma(4, πœƒ)$, and let $T = \frac{1}{4} \overline{X}$ be an estimator of $πœƒ$. Determine: a) $E(T)$ b) $Bias(T;πœƒ)$ c) $MSE(T;πœƒ)$ d) whether $T$ is an MSE-...
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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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Converting expectation to probability for generative model

I'm dealing with a problem in which I have many individual samples, and each of them has a non-negative numeric outcome. I want to predict, for groups of these samples, the expected value of the sum ...
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E(2x-y)^2 in mathematical expectation [closed]

X and y are independent variable and information is as follow: E(x)=4 E(y)=6 V(X)=5 and V(y)=4 here E stands for expectation and V for variance. What will be the value of E(2x-y)^2 ? What will be ...
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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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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
27 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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24 views

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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1answer
37 views

Expectation of 1/x when x is discrete distribution including 0

I'm trying to understand what the expectation of $1/X$ would be when $X$ can take on values of $0$. I've looked this up, and I understand that for continuous distributions you can take limits. However,...
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36 views

Student-t Fisher Information wrt log(sigma)?

I am having some issues finding any information on how to compute the fisher information of a student-t distribution when the standard deviation is parameterized as $log(\sigma)$ rather than $\sigma$. ...
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expected number of dependent trials until n successes

In case of bernoulli experiments, I know that the expected value for the number of trials needed to have $n$ successes is n/p. (where p is probability of success for each trial). Now my question is, ...
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Question about true risk and empirical risk of sample S, when both are hypothesised by ERM of S

I was reading through some notes online, and I came upon a property that I don't know how to prove. This is the property: E [ R(ERM(D)) - R'(ERM(D), D) ] >= 0 where D is a sample, ERM(D) is the ...
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Expected squared distance between order statistics?

Suppose $p(\cdot)$ is a smooth probability distribution over $\mathbb R$. Suppose we draw two collections of $k$ i.i.d. samples from $p(\cdot)$, yielding random variables $(X_1,\ldots,X_k)$ and $(Y_1,...
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Demonstrating a form of E(SSR) via quadratic forms

I've been given a problem asking that I show that, in a normal error regression model, $$E(SSR)=Οƒ^2+Ξ²'X'(I_n-(1/n) J_n)XΞ²$$ I am new to applying matrix algebra to statistics, but I do know that the ...
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8 views

Find range of values 95% of the time for a weighted sample

I want to generate some data about a new game I am playing. There exists a mechanic in the game where you can critically strike, dealing extra damage. You can also deal different amounts of damage in ...
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1answer
38 views

basic questions about expected value [closed]

I'm trying to learn machine learning and I'm filling in the gaps in my knowledge as I go along. I see from this definition that $$ E[X] = \int_{\mathbb{R}} xf(x) dx $$ But what is $E[\hat{\beta}|X]$? ...
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Find expectation of conditional normal distribution

I am struggling with some finding expectation value question . the question is to find $E[Y|X]$ from the result $P(Y|X)$ with given mean and covariance $$\mu=[\mu_x, \mu_y]^T$$ $$\Sigma=\begin{bmatrix}...
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Calculation of $E\left[\frac{X}{E[X]}\right]$

how can I rewrite $E\left[\frac{X}{E[X]}\right]$? I did: $E\left[\frac{X}{E[X]}\right] = E[X] * E\left[\frac{1}{E[X]}\right] + \operatorname{cov}\left(X, \frac{1}{E[X]}\right)$ How can I continue? Is $...
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1answer
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Showing that a discrete random variable has the same moments as a Normal Distribution

Suppose I define $X$ to be normally distributed with $\mu = 0, \sigma^2 = 1$, so that $X$ has the pdf $f_{X}(x) = \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2}, \quad -\infty < x <\infty.$ Let discrete ...
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Interpretation of $|cor(X,Y)|$ as a common variance and generalisation to a bigger number of variables

Assuming that $X, Y$ are standardized random variables, can we interpret value $|E[XY]|$ as a proportion of "common/shared" variance between $X$ and $Y$? If yes, then if $Z$ is a ...
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1answer
24 views

Bounds for the expectation of the max

Consider the following expected value $$ E(\max\{a_1+\epsilon_1, a_2+\epsilon_2,...,a_n+\epsilon_n\}) $$ where the expectation is taken with respect to the random variables $(\epsilon_1,...,\epsilon_n)...
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Expectation of a Function of a RV Problem

Problem. Let $X$ be a random variable (either continuous or discrete) that takes nonnegative values. Prove or provide a counterexample to the following statement: There does not exist an X such that $...
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How do you write the expected value of an arbitrary random variable $X$ in terms of $F_X$?

The "Darth Vader rule" for the expected value of non-negative random variable is: $$\mathbb{E}(X) = \int \limits_0^\infty (1-F_X(x)) \ dx.$$ This rule applies only to non-negative random ...
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How can I mathematically prove this time series when $e_t$ has i.i.d distribution?

Not really sure on how to simplify the $y_t$ because there is $y_{t-1}^2$ In order for a time series to be Martingale difference sequence the expected value given all the past value should be 0 and ...
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1answer
47 views

Expected Value of $Y/X$ [closed]

Consider two random variable, $X$ and $Y$ such that $E(Y\mid X)=0.5X$ and $E(Y) = 20$ and $E(X) = 10$. Compute $E(Y/X)$.
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Expected number of trials to get 5 heads [duplicate]

Say I have a fair coin. I want to know the expected number of trials to get 5 heads. What is the relationship to the following idea: $E[$number of heads in n trials$]= (1/2 \cdot 1+1/2 \cdot 0) \cdot ...
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Expected Value of Total Number of Three Consecutive Colored Balls

Suppose we have 52 balls in random order where 26 are colored blue and 26 are colored red. The total count of a triplet of red balls in order or whenever a red ball directly follows two previous red ...
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Expected value of unbiased estimator of $\sigma$ in binomial sum

Suppose that $Y_1, Y_2, \dots, Y_r$ are random independent variable such as $Y_i \sim B(m_i, \pi)$, the idea first is to find $\hat{\pi}$ which is the maximum likelihood estimator an use it to find ...
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Probability hotel reservations

A hotel has 100 rooms, and charge guests for their rooms in advance. The number of reservations for tomorrow night is denoted as $n$. Rooms are held until 10pm, but if a guest hasn't shown up by 10pm, ...
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Can you explain the underlying math in this question please? Regarding continuous expected values

$X$ is a random variable. If I am given the $P(X<1)= 0$ and the $P(X>e) = 0$ and in the range $y$ in $[1,e]$ the $P(X<y)= \int_1^y \frac{1}{x} dx$. If $\mu$ is the expected value of $X$, what ...
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1answer
36 views

same cdf equals same expectation?

So, if $X$ and $Y$ are both continuous random variables with the same cdf, does that mean that their expectations are the same? And the same thing in case $X$ and $Y$ are both discrete. Thanks in ...
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1answer
37 views

What is the expected value of the number of experiments required to obtain the first success?

In a simulation experiment, two independent observations $X_1$ and $X_2$ are generated from the Poisson distribution with mean 1. The experiment is said to be successful if $X_1 + X_2$ is odd. What is ...
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34 views

what is the expected value of the dot product of two vectors

I have a little question, but I don't know that well how to answer it. I have a random walker with position vector $\vec{r} = \sum_{i=1}^N \vec{r}_i$, where i is the random walker's step. Every vector ...
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1answer
21 views

Formula of Expected Shortfall for Generalized extreme value distribution (GEV)

i found the formula ofthe ES for GEV here: https://en.wikipedia.org/wiki/Expected_shortfall#Generalized_extreme_value_distribution_(GEV) My problem is that there is no citation, but I need the formula ...

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