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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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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The expected value and variance of E(-1X)? [closed]

This might be a stupid question, but how I can calculate the expected value $\operatorname{E}(-1X)$ and variance $\operatorname{Var}(-1X)$ for example in a case in which $X\sim N(100,0.1^2)$?
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Likelihood of my friend being able to guess skittle taste

I'm preparing for a data science interview, and here's a question I encountered during my preparation: Your friend claims he can tell the five colors of skittles apart by taste alone. The probability ...
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1answer
33 views

Understanding covariance

I came across following problem: A discrete random variable $P$ takes values $-3,-2,0,2,3$ with probability $0.2$. Let $Q=P^2$ be another random variable. What is covariance of $P$ and $Q$? I solved ...
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Covariance for Random Variables vs. Sample Data

In my textbook, it says that the formula for finding covariance between two random variables is: $Cov(X,Y)=E((X-EX)(Y-EY))$ With $EY$ and $EX$ being the mathematical expectation for the random ...
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Expectation of a sequence of random variables based on a set of iid Gaussian random variables

This is a rather convoluted problem: I'll my best trying to explain it. So, we have $m$ iid standard Gaussian RVs $Q_i$. We get a realization from each of them, and these values $q_1,\dots,q_m$ are ...
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Help me decipher the notation in the given definition

I was going through a book on statistics and I came across this definition of expectation. Let $X : \Omega \rightarrow D $, be a discrete random variable and let $\phi : D \rightarrow {\rm I\!R}$ The ...
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Expectation of truncated autoregressive process

Consider the following nonlinear autoregressive model: \begin{align*} \epsilon_t & = \epsilon^\rho_{t-1} e^{u_t}, \\ x_t & = \frac{\gamma \, \epsilon_t x^a_{t-1}}{1+b y_t}, \\ y_t & = \...
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How to derive the expectation of $\ln \mu_j$ in Dirichlet distribution

I have derived the mean and variance of $\mu_j$ in Dirichlet distribution $\text{Dir}(\mu_1, \cdots, \mu_K|\alpha_1, \cdots, \alpha_k)$. On https://en.wikipedia.org/wiki/Dirichlet_distribution, it ...
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Upperbound for Norm of Lagged Autocovariance Matrix

Suppose $X_t$ is a vector valued time series in $\mathbb{R}^d$. Assume for the moment that $X_t$ is stationary with $EX_t=0$, $E\|X_t\|^2<\infty$, and let $$ C_h = E[X_0 X_h^\top] $$ denote the ...
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Estimating Population Total of a Lognormal distribution

Say we’re trying to model spending behavior and it has a lognormal distribution, lognormal(6.4, 0.8) with N=1000 independent observations, a vector named A. What’s the expected value of the total ...
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Relationship between two “loss-like” variables

Suppose I have a discrete set $A \subset \lbrace \theta\in\mathbb{R}^3 : \lvert\lvert\theta\rvert\rvert_2 = 1 \rbrace$ of unit vectors, and let $F: A\times \mathbb{R}^n \to R^k$ such that $F(\cdot ,C)$...
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How do you calculate the expected value of $E\left\{e^{-|X|}\right\}$ e.g. for Gaussian X?

If $X$ is a random variable, I would like to be able to calculate something like $$E\left\{e^{-|X|}\right\}$$ How can I do this, e.g., for a normally distributed $X$?
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220 views

Difference between the expectation of x bar squared and the expectation of x squared

I am trying to understand the derivation of the expectation of the maximum likelihood (MLE) of variance, however I am confused as to what the difference is between $\bar{x}$ and $x$. Below you find ...
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expected value of a fishing strategy

Suppose there is a pond with infinite number of fish. Weights of the fish are iid uniform $(0,1)$. We catch fish from this pond with the following rules: Each day we catch at most one fish from the ...
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Expectation and variance of quotient of sums of positive, discrete, iid random variables

Let $\{X_i\}_{i=1}^n$ be $n$ positive, discrete (so positive integers) and IID random variables. Let $\{c_i\}_{i=1}^n$ be constants and $$Y=\frac{\sum c_iX_i}{\big(\sum X_i\big)^2}\ \ \ ;\ \ \ Z=\frac{...
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Weighted Conditional Expectation definition in AdaBoost

I am looking at "Additive logistic regression a statistical view of boosting" paper (https://web.stanford.edu/~hastie/Papers/AdditiveLogisticRegression/alr.pdf) In page 346, the authors ...
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How do you get the double sum or integral from $E(X+Y)$ (expected value)?

I was given a proof for $E(X+Y)$ = $E(X)+E(Y)$ for cases where both variables are either discrete or continuous: Discrete: $$ \begin{align*} E(X+Y) &=\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}(x+y)...
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Upper Bound and Lower Bound on Means when Distributions are bounded?

Suppose we have two different probability distributions $p, q$ defined on input $x \in [0,1]$. We know that for any value of $x$ in the domain, we have $\exp^{-a} \leq \frac{p(x)}{q(x)} \leq \exp^{a} $...
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Intuition behind unbiased OLS estimator derivation

I was going through the derivation of unbiased OLS estimator $$E(\hat{\beta_1}) = \beta_1 + (1/SST_x) \sum_{i=1}^n d_i E(u_i) = \beta_1 + (1/SST_x) \sum_{i=1}^n d_i\cdot 0 = \beta_1$$ My doubt is if $...
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Number of Head-Tail in N independent coin flips

Consider a sequence of independent tosses of a biased coin at times $t = 0, 1, 2, . . .$. On each toss, the probability of a ’head’ is $p$, and the probability of a ’tail’ is $1 − p$. A reward of one ...
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Taking Expectation Over Inverse Sum of Indicator Functions?

I'm working with a zero inflated Poisson distribution that has the following pmf: $$f(y|w,\lambda)=wI[y=0]+(1-w)\frac{e^{-\lambda}\lambda^{y}}{y!}$$ I would like to find the expectation of the ...
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Expected value of indicator variables

I have an ensemble of graphs with $N$ nodes and $M$ undirected edges each. Let $X_{ij}$ be indicator variables that point to the existence of an edge between nodes $i$ and $j$ on a particular graph. ...
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Minimize $\mathrm{E}(U-bV-a)^{2}$, $U=\sum_{i=1}^{3}X_{i}.\, V=\sum_{i=1}^{3}i X_{i}\quad X_{1},X_{2},X_{3}\in N(1,1)$ i.i.d. r.v

Let $X_{1}, X_{2}$ and $X_{3}$ be independent , $N(1,1)$-distributed random variables. Set $U=X_{1}+X_{2}+X_{3}$ and $V=X_{1}+2X_{2}+3X_{3}$. Determine the constants $a$ and $b$ so that $\mathrm{E}(U-...
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Mean and Variance of dot product of 2 random vectors?

x and y are two vectors of dimension k. Assume that the components of x and y are independent random variables with mean 0 and variance 1. What would be the mean and variance of their dot product, x · ...
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$E(xy)<\infty$ proof

I am reviewing the best linear projection properties proof in Hansen's book on econometrics. Specifically, the proof according to which $E(xy)<\infty$. For this, it is assumed that $E(y^2)<\...
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Bias/variance of an estimator versus bias/variance of a model

I have come across two different but similar definitions of bias and variance. In the book Deep Learning book, in section 5.4, bias and variance are defined for an estimator, where bias is the ...
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Proof that Cov(W+Y, Y-V) = 0 given that W, Y, and V are uncorrelated but not independent

$\newcommand{\Cov}{\operatorname{Cov}}$ I'm trying to prove the following statement: $\Cov(W+Y , Y-V) = 0$, given the following constraints: $W$,$Y$, and $V$ are Uncorrelated but not independent $E(W)...
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Counterexample where E(u|x)=0 in a regression model cannot hold in the population?

Edit: Background information: I have two variables of interest, $y$ and $x$ that are linearly related via the following: $y = a + bx + u$, where "$a$" and "$b$" are fixed ...
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Find $E[N^2 | N > 2]$ for a frequency distribution

N has probability mass function: $p_o = p_1 =0$ and $p_k = \frac{1}{(e^1-2)k!}$ for $k=2,3,4,...$ I Solved for the pgf of N and got $G(t) = \frac{e^t}{e^1-2}$ How do I calculate $E[N^2 | N>2]$?
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What is the following expectation?

What is $E[Y_1I(x<Y_2,Y_2>Y_1)]$, where $Y_1$ and $Y_2$ are non-negative continuous random variables and x is a constant? $I(.)$ is the indicator function. Can this be written as follows? \begin{...
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Doubt in derivation of expectation of sample variance

I am studying statistics on my own. Please help me in understanding following Here in evaluation of expectation $E[\frac{(n-1)S^2}{\sigma^2}]$, why $\sigma^2$(population variance) is treated as ...
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Exogeneity assumption applied to functions of the design matrix

The context of this question is ordinary least squares. $X$ denotes the design matrix. I would like a proof of the claim – or a corrected version thereof – made in this other question that the ...
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How do derive the expected prediction error for OLS?

At the end of section 3.2.2 of Elements of Statistical Learning, it shows the following: I am having a hard time deriving this. This is what I have so far: \begin{align} E[(Y_0 - \hat{f}(x_0))^2] ...
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Intuition for expectation of discrete random variable that takes positive integers

If $X$ is a discrete random variable that takes values on the positive integers, it is true that $$E(X) = \sum_{k=1}^{\infty} P(X \ge k)\;.$$ I know how to prove this (by expressing the summand as a ...
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Doubt in independence of 2 random variables

If 2 random variables are independent, then $f(x,y) = f(x)f(y)$. Is converse true? $F(x,y)=F(x)F(y)$. Is converse true? $E(x,y)=E(x)E(y)$. Is converse true? where $F$ is cdf and $f$ is pdf I recently ...
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Expectation of a absolute of centered RV is bounded by root of variance?

I've seen a claim that for a RV x where $E(X)=0$, it holds that $E(|X|)\le\sqrt{Var(X)}$. How can you prove this? (and maybe what is the intuition for this?) Thanks
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Show that if $Y$ is another random variable such that $E[X] = E[Y]$ and $V(X) = V(Y)$ then $P(Y \ge a) \le p$

Let $p \in (0,1)$ and $X$ be a random variable such that $P(X=a) = p, P(X=-b) = 1-p$ Show that if $Y$ is another random variable such that $E[X] = E[Y]$ and $V(X) = V(Y)$ then $P(Y \ge a) \le p$ and ...
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How many trials are needed to have 99% of all possible successes between n independent elements which can have at most 2 successes each?

I want to calculate the expected time of growth of a group of plants in a videogame. If I have $n$ independent plants and each plant has a probability $p$ of being picked each unit of time $t$ (or ...
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Calculating the expected value of the log multinomial probit

I am stuck trying to solve the calculation of an expectation related to the multinomial probit likelihood. Say I have a random vector $\mathbf{F}$ whose components, $F_1$, $F_2$, $F_3$, are ...
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Law of unconscious statistician for conditional expectation

I have random variables $X$, $Y$ with joint distribution $f_{XY}(x,y)$ and conditional distribution $f_{X|Y}(x|y)$ and another random variable $Z=g(X)$ with $g$ being bijective is it true that $$E(Z|Y=...
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Decomposition of the second moment of a circular distribution

When we have a "usual" random variable $X$ on the real line, we can break down the second moment. $$ \mathbb{E}[X^2] = (\mathbb{E}[X])^2 + var(X) $$ Is there an equivalent for a circular ...
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Where does the expected value definition come from? [duplicate]

The definition of the expected value on the domain $[a,b]$ is given by $$E[X] := \int_a^b x f(x) \, \mathrm dx $$ I understand what the mean is, but I don't fully understand how this specific equation ...
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Comparing performance to random in multiple choice test

Background TLDR I was asked to grade some multiple choice tests. I noticed a difference between two groups (which differ by the part of our online materials they used). But I think it would be more ...
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What is the expectation of absolute difference

Given: E(|X-R|) =2.5 E(|L-X|)= 3 where X, R, L are assumed to be normally distributed and independent. Standard deviations of |X-R| and |L-X| are known. what is E(|L-R|) ?
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488 views

Expected value of X^3 for a normal distribution given it has limits?

What is the expected value of $X^3$ with in limits for a normal distribution? In other words, I am looking for solution of $E(X^3 \mid a\leq X \leq b)$.
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Laplace and Normal Distribution Cross Entropy

I need the following integral and struggle with calculating it or finding a citable source. $$\int_{-\infty}^{\infty}(x-\mu)^2\exp\!\left(-\frac{|x-\nu|}{\tau}\right)dx.$$ Background: I want to find ...
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Creating an expected dataset to compare with observed data to test for genetic interactions

I am trying to define a test to tell if two mutations combined give a greater phenotype than what you would expect by simply adding both. Here is an example (using R): ...

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