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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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, $$E(SSR)=σ^2+β'X'(I_n-(1/n) J_n)Xβ$$ I am new to applying matrix algebra to statistics, but I do know that the ...
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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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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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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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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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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 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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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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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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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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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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Bias in Gradient Descent (GD) and Stochastic GD (SGD)

Let $\theta$ be weight parameters and assume the loss function to be $L_N(\theta)=\frac{1}{N}\sum_i f(\theta; x_i,y_i)$. Assume a mini batch loss function with a batch of size $M$ and denote the loss ...
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Expected value of a random variable by integrating $1-CDF$ when lower limit $a\neq 0$?

I have found several past answers on stack exchange (Find expected value using CDF) which explains why the expected value of a random variable as such: $$ E(X)=\int_{0}^{\infty}(1−F_X(x))\,\mathrm dx $...
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What does $E(X = k)$ mean if $X \sim Geom(p)$?

So, I have the following question for homework assignment: $X_{1}$, $X_{2}$ have geometric distributions with parameters $p$ and $1-p$ respectively, $X_{1} \sim Geom(p)$, $X_{2} \sim Geom(1-p)$. $X_{1}...
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Expected number of uniform draws to exceed a first uniform draw

I came across the following problem (Problem number 27 from here): Aaron samples from the Uniform(0,1) distribution. Then Brooke repeatedly samples from the same distribution until she obtains a ...
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Monte Carlo standard error for a sum

Suppose that I want to compute $E[X+Y]$ using Monte Carlo simulation and compute the standard error. (Note: $X,Y$ are not necessarily independent) The standard way to do this is to Consider the ...
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Expectation of most occuring face of tossing a random dice multiple times

Given a biased dice (having $m$ faces) and toss it $n$-times and report the count of the most occurring face (denoting as $R$). The degree of bias of the dice is random. What is the expectation value ...
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Question about an implication relationship

Let $||\cdot||$ be the Euclidean norm. Suppose $X_1,X_2$ are two independent and identically distributed random variables, and $a_N(X_1,X_2)$ is a vector valued function that depends on factor $N$ and ...
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What does it mean to take the expectation with respect to a probability distribution?

I see this expectation in a lot of machine learning literature: $$\mathbb{E}_{p(\mathbf{x};\mathbf{\theta})}[f(\mathbf{x};\mathbf{\phi})] = \int p(\mathbf{x};\mathbf{\theta}) f(\mathbf{x};\mathbf{\phi}...
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With high probability analysis

I am quite confused with 'with high probability analysis' in randomized algorithms. Perhaps it's helpful to illustrate with an example. What is the expected number of rounds to have at least one head ...
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Difference between Sample Mean($\bar{X}$) and Expected Value (E[X])

I know this has been asked and answered, but many even have contradictory definitions as to which represents the mean of a probability distribution. I would like to cite: https://stats.stackexchange....
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Expected value of random variables that are generated by substraction from the mean [duplicate]

If suppose we have $ X_1,X_2,\ldots,X_N$ that are independent Normal random variables with mean $\mu$ and variance $\sigma^2$, $X\sim N(\mu,\sigma^2)$. And if we have $Y=\mu-X,$ then is the mean of $Y$...
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1answer
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Expected vallue calculation of i.i.d. random variables

Suppose $X_1,X_2,\ldots,X_n$ are a sequence of i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Define the sample mean $\bar{X} := \frac{1}{n} \sum_{i=1}^{n} X_i$, which we know is an ...
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Unbiased estimator for a parameter from a transformed distribution

I am solving an exercise in which I have to show that a certain estimator is unbiased for a given parameter. However, after a couple lines of computation I got stuck in the following scenario: $$ \...
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1answer
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Can't Follow the Algebra in a Estimator MSE Comparison

Little bit of background - working through some maths and stats autodidactically. I simply can not follow the algebra of the following worked example comparing the MSE of two estimators. I can not ...
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Expected number of running heads in coin toss

How to find the expected number of running heads of a specific length (say 'k' exactly) in 'n' tosses of a coin (fair/biased). For example, consider the output of a coin toss as follows "...
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Expectation of log skew normal distribution

What is the expected value and expected variance of a log skew normal distribution? In case I have the terminology wrong, I'm referring to data that is lognormal with some skew mild skew when it's log ...
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$EY$ in a linear regression

Let $y=x^T\beta+e$ be a linear regresion with $e$~ $N(0,I\sigma^2)$ What can we say about the expected value of $y$ $(EY)$? Is it correct $EY=x^T\beta$?
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Expectation of a type of bivariate lognormal

Suppose $S_1 \sim e^\mathbf{X}$ where $\mathbf{X} \sim N(\mu, \mathbf{\Sigma})$, $\mathbf{X}$ is a bivariate normal distribution then what is the following, $$ E\left[ \theta_1^\intercal S_1 \right] $$...
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If the predicted value of machine learning method is E(y | x), why bother with different cost functions for y | x?

Say we know that $Y$ follows a distribution with density $f$. It is well known that the mean $E(Y \ | X)$ minimizes the Root Mean Square Error (RMSE). We are generally interested in predicting a value ...
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What is the variance of an integral random variable?

In particular i have to find the variance of this random variable $$U = \int_{-\infty}^{y} \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(x-T)^2} dx$$ where T is a ranadom variable distributed with a normal ...
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What is the fisher information matrix of the multivariate t distribution?

$\newcommand{\bx}{\mathbf{x}}$ $\newcommand{\bSigma}{\boldsymbol{\Sigma}}$ $\newcommand{\bE}{\mathbf{E}}$ $\newcommand{\bD}{\mathbf{D}}$ Consider the multivariate central t distribution with p.d.f. \...
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Expected value of the largest item in a multinomial distribution

In this question, I use the notation on wikipedia. Suppose $X=(X_1,\ldots, X_k)$ follows a multinomial distribution with parameter $n,\mathbf p$, where $n$ is the sample size or the number of trials, ...
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Why is the answer to this recurrence relation this?

(Cross-posted from math-SE for visibility) self studying some probability theory and found this problem: Michael Banks randomly selects cubes from Mary Poppins' magical bag (an infinite urn). In each ...
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What is a random variable and what isn't in regression models

I've already seen this question but it didn't help . So I'm going over regression models (simple linear regression mainly) in my statistics text book and there's a lot of confusion here about what ...
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Graphical representation of unconditional expected value

First, let tell you that I've being struggling with the concept of unconditional expectation for linear regression. For conditional expectation is easier: We know that the conditional expectation ...
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Expected value is known

I'm a agronomy student in Colombia and I've been recently studying from the book Generalized Linear Models With Examples in R by Dunn and Smyth. As you can imagine, I do not have a pretty good ...
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Probability that number of heads exceeds sum of die rolls

Let $X$ denote the sum of dots we see in $100$ die rolls, and let $Y$ denote the number of heads in $600$ coin flips. How can I compute $P(X > Y)?$ Intuitively, I don't think there's a nice way to ...

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