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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Expectation of a random walk that can't go below zero

Suppose we have a random walk $S_n$ that is constrained to be positive or zero, that is: $$S_0 > 0$$ $$S_{i+1} = \max(S_i+x_i,\space 0)$$ $$x_i \sim N[\mu,\sigma^2]$$ Can we analytically ...
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Finding the Cauchy Principal Value Mean of the pdf for $Z=XY$ where $\mathbb{E}[Y]$ does not exist

Let's first define the Cauchy Principal Value Mean (PVM). For a continuous random variable $V$ with pdf $f_{V}(v)$, the PVM of $f_{V}$ is \begin{equation} \mathrm{PV}(\mathbb{E}[V])=\lim_{a\to\infty}\...
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804 views

Expectation of a strictly increasing function

Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
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937 views

Conditional expectations by conditioning on functions of random variables

I have conjectured the following: Let $f:\mathbb{R}\supseteq A \rightarrow B \subseteq \mathbb{R}$ be an injective function. Let $X$ be a random variable with support $A$ and $Y$ be some random ...
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Is the expected value a valid decision-making method in a very short term?

This might be related to game theory more than statistics, but I decided to ask this question here. Let's assume you're offered a lottery. There are a hundred balls in a bowl: 99 white balls and one ...
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36 views

Why is the observed Fisher information defined as the Hessian of the log-likelihood?

In an MLE setting with probability density function $f(X, \theta)$, the (expected) Fisher information is usually defined as the covariance matrix of the fisher score, i.e. $$ I(\theta) = E_\theta \...
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57 views

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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214 views

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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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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90 views

Calculating the variance of a weight-average when the weights also have a variance

Assume there is a series of random variables $X_1$, $X_2$, ..., $X_N$ representing a series of values to be weight-averaged, and a corresponding series of random variables $W_1$, $W_2$, ..., $W_N$ ...
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179 views

Taylor Series Expansion of Unconditional Expectation

We know that the best 1st order approximation of an unconditional expectation is the following- $$E(y|x)=(E(y)-\beta E(x))+\beta x$$ where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}...
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738 views

Concentration of maximum of subexponential random variables

I'm looking for a concentration bound on the maximum of a collection of sub-exponential random variables, which are not necessarily independent. More specifically, I have the following collection: \...
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291 views

Expectation of log of shifted LogNormal random variable

Suppose that we have a shifted log normal random variable, $$Y = X + \beta, \qquad X \sim \log \mathcal{N}(\mu, \sigma^2), \qquad \beta > 0.$$ I'm struggling to obtain $\mathbb{E}[\log(Y)]$ Do I ...
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Expected value of a “logistic uniform” multivariate

Let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb R^d$ and $b_1,\ldots,b_n \in \mathbb R$ be fixed. For $\mathbf{x} \sim \mathcal U([0,1]^d)$ and $j \in \{1,\ldots,n\}$, consider the real variable ...
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Expected value of softmax transformation of Gaussian random vector

Let $\mathbf w_1,\mathbf w_2,\ldots,\mathbf w_n \in \mathbb R^p$ and $\mathbf v \in \mathbb R^n$ be fixed vectors, and $\mathbf x \sim \mathcal N_p(\boldsymbol{\mu}, \mathbf{\Sigma})$ be an $p$-...
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Calculate expected value of CDF of a different Beta variable

Let $$ X_1 \sim Beta(\alpha_1,\beta_1) \\ X_2 \sim Beta(\alpha_2,\beta_2). $$ Let $F_X(x) = P( X \le x )$ be the CDF of $X$ and $\mathbb{E}_{X}(\cdot)$ be expectation with respect to $X$. How to ...
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When does $\forall p: E_p[f] = E_p[g]$ imply $f = g$?

Let's say $$E_p[f(X)] = E_p[g(X)]$$ for all $p \in S$, where $S$ might be some parametric family of densities, for instance. Under which assumptions on $S$ does this imply $f = g$? I am reading Efron ...
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Interesting application of E-M algorithm

Suppose the following dataset: [3 4 3 4 6 12 12 7 8 9] [2 5 3 4 12 2 2 10 7 6] [3 4 3 4 5 11 10 7 8 9] These numbers are totally random. So this dataset, depicts ...
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Problem with “Pattern Recognition and Machine Learning” (Bishop) Ch. 10

I'm working on putting together a variational solution to a somewhat "exotic" mixture model, and got stumped setting up the computation for the lower bound. This prompted me to go back to this ...
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Expected value of $\frac{\overline X}{1-\overline X}$ when $X_i$'s are i.i.d $\mathsf{Beta}(\theta,1)$

I am trying to determine $E\left[\frac{\overline X}{1-\overline X}\right]$, where distribution of $X_1,\ldots,X_n$ is $$f(x;\theta)=\theta x^{\theta−1}\quad,\, 0 < x < 1\,,\, \theta > 0 $$ ...
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Distribution of ratio of 2 points drawn from normal distribution?

Let's say we have a known normal distribution $N(\mu,\sigma^2)$. I now draw 2 points $p1$ and $p2$ randomly from this Gaussian distribution for every observation, and repeat this process large number ...
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Deeper proof of common expected value of random convex weights

$\newcommand{\E}{\mathbb{E}}$Let a finite collection of exchangeable random variables $X_1,...X_n$ (some authors would call this collection "interchangeable" since it is finite, reserving "...
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Modelling of probabilistic vs deterministic systems

The learning problem in Statistical Learning Theory is defined as: $$ R(f) = \int_{X,Y} L(y, f(x))P(x,y)\mathrm{d}x\mathrm{d}y $$ where $R(f)$ is the expected risk $L$ is the loss function $P(x, y)...
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How do I solve $E\left[ E \left(X|Z \right) E\left( Y|Z \right)\right]$?

I am trying to solve $E\left[ E \left( \mathbf{X}|\mathbf{Z} \right) E \left( \mathbf{Y}|\mathbf{Z} \right) \right]$, (where $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ are random variables) but I am ...
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276 views

Expectation of density ratio of two iid variables

Let $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent RVs and let $f$ be their density function. I'd like to compute the expectation of the density ratio \begin{align} \mathbb{E}\left[\frac{f(X)}{f(Y)...
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An inequality involving expectation

Let $f,g$ be two pdfs, and suppose $X$ is a random variable that has pdf $f$. Is it necessarily true that $E[f(X)] \ge E[g(X)]$? Although I doubt this will help, but I got this problem from studying ...
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Smooth expectations outside the exponential family

At page 85-86 of Young and Smith "Essentials of Statistical Inference" there is an interesting result. If $X$ is a r.v. distributed according to the exponential family and $\phi(x)$ is a bounded (but ...
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Minimizing $L^p$ distance of a random variable and a constant

Let $X$ be a real-valued random variable. Then expected value $\mathbb{E}[X]$ is the number $c$ minimizing $c \mapsto \mathbb{E}[(X-c)^2]$. Similarly, the median of $X$ is the number $c$ minimizing $c ...
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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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47 views

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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66 views

Rigorous statement of expectations for the bias-variance trade-off

Consider a data generating process $$Y=f(X)+\varepsilon$$ where $\varepsilon$ is independent of $x$ with $\mathbb E(\varepsilon)=0$ and $\text{Var}(\varepsilon)=\sigma^2_\varepsilon$. According to ...
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The expected value of $\frac{1}{\sqrt{1-r}}$ where $r$ is Pearson correlation

I am looking to unbias the sample statistic $\frac{1}{\sqrt{1-r}}$ where $r$ is a Pearson correlation. The population is assumued binormal with equal variance $\sigma$ and with true correlation $\rho$....
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How do I calculate the Expected value of this normal distribution?

I have a distribution Yi - BXi ~ N(0,T) and I'm having trouble calculating an element of the Expected Information matrix for it. The density function should be f(Y, X; B, T) = $\frac{1}{\sqrt{2πT}}$ ...
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122 views

Expectation of kth order statistic of Pareto distribution

I am trying to find the expected value of $X_{(k)}$ Given cdf $$ F(x) = \begin{cases} 1-\left(\frac{\sigma}{x}\right)^\alpha, & x > \sigma\\ 0, & \text{else.} \end{cases}$$ My attempt: $$...
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Expected value of quotient of Poisson distributions

Let $X$ and $Y$ be independent random variables such that $X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$. Is there ...
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Multiple interval ratio of E[X / (X + Y)]

I have a sequence of interchanging on- and off-intervals, each pair identified by index $i$. The duration of the on-interval $i$ is represented by random variable $X_i$, and the duration of the off-...
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How to approach Basketball “Beat the Pro” drill with Markov Chain

Suppose, similarly to Gambler's ruin problem a Basketball player is doing "Beat the Pro" drill. That is, for every shot made, he scores one point, and for those missed, two points are to be deducted. ...
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Expected eigenvalues of a Wishart Matrix

I consider a $n\times n$ Wishart Matrix with expected value $p \cdot I_n$, i.e. a matrix of the form $$W = XX'$$ with $X$ a $n\times p$ matrix with independent standard normal entries. It is easy ...
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Mean of a Pearson Type VI distribution

I have a question that gives me the density of a Pearson type VI distribution and then says to state the range of parameter(s) for which the expression for the mean is valid. In your calculations, ...
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394 views

expectation involving the normal CDF

I'm looking for a closed form expression for $$\mathbb{E}\left[\Phi\left(aZ+b\right)^{k}\right] = \int_{-\infty}^{\infty}\Phi\left(az+b\right)^{k}\phi(z)\,dz$$ where $a$ and $b$ are real numbers, $k&...
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121 views

Expected Value of this game

I've got a game that can be described by flipping coins over 7 days. Every day I get a number of coins described by a Poisson distribution. Each coin flips heads with probability $p$, and if I succeed,...
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423 views

Expected value of $\sigma$ in GARCH model

In a standard GARCH(1,1) process defined as: $$\varepsilon_t=\sigma_t z_t \\ \text{ where } z_t \sim \text{ iid } N(0,1) \\ \text{ and } \sigma_t^2 =\alpha_0 + \alpha_1 \varepsilon^2_{t-1} + \alpha_2 ...
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250 views

Expectation of roots of a quadratic equation

The quadratic equation $x^2 -ax+ b = 0$ is known to have two real roots, $X_1$ and $X_2$ $(X_1 > X_2)$ but the coefficient $b$ is a positive unknown and can be assumed to have a uniform ...
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466 views

Expectation of the log of a ratio of two lognormal random variables with additive constants

I have two independent lognormal random variables $X$ and $Y$ with known means and variances. I would like to know the expected value and the variance of $\ln\big((X+1)/(Y+1)\big)$. If there is no ...
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On proving that a random variable is a conditional expectation

The definition of conditional expectation I am given and want to work with in the following proof is: Let $X \in L(\Omega, P) $ on $(\Omega, F, P)$ and let $G \subset F $ be an other sigma algebra ...
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Determine if a series of discrete distributions are expected

*I apologize for the length of this post and I have almost no statistics experience, please keep that in mind :) In competitive diving, a diver will perform 5 different dives and will receive scores ...
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216 views

On the expectation of the reciprocal of a quadratic form of a standard normal variable

$\newcommand{\Tr}{\operatorname{Tr}}$We know that if $x$ is a p-variate standard normal random variable, then $x^T A x$ converges to the $\Tr(A)$. Reference. Suppose that $p>2$. Question: What if ...
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Cumulative count of events

Consider an active chat room which has less than $N$ messages so far, each with a timestamp. How can I determine the first day where the expected cumulative message count has a $k$ chance of ...
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432 views

Expectation of log(1/X) when X follows inverse gaussian distribution

Can anybody help me in this question? How to derive the expectation of log(1/X) when X follows inverse Gaussian distribution? Found out a type of approximation for $\mathbb{E}[\log(X)]$ while $X \sim ...
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A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$

I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...

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