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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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Trying to Understand Power Series Expansion of Cumulant Generating Function

I'm having difficulty expanding the cumulant generating function, $K_X(t)$ as a power series. Specifically, I'm trying to go from the definition of the cumulant generating function, i.e. $$K_X(t) = \...
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For any three random variables $X,Y,Z$, prove or disprove that $\langle XY\rangle$, $\langle XZ\rangle$, $\langle YZ\rangle$ can't all be negative [duplicate]

For any three real random variables $X,Y,Z$, prove or disprove that $\langle XY\rangle$, $\langle XZ\rangle$, $\langle YZ\rangle$ can't all be negative. Here $\langle \cdot \rangle$ denotes an ...
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Can additional iterations of backward induction as described affect optimal policy?

Consider a game with the following properties: Single player Finite number of game states (after the player arrives at a terminal state, he or she can begin again from the start state; the player can ...
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Expectation of random sum of dependant variables

The expectation of random sum of independent identically distributed variables is given either by the law of total expectation or by Wald's identity. Are these generalised to tackle the random sum of ...
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20 views

Expectation Help

I am having trouble a question in my probability course and was looking for some insight into it. A digital clock sits next to your bed and in the mornings when you don't use an alarm you notice Y =...
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Finding OLS estimator for $\beta$ where $y_i=\beta+ 2 \beta x_i+\epsilon_i$

Consider the following model with the usual OLS assumptions: $\epsilon_i$ are uncorrelated random variables with mean zero and constant variance $\sigma^2$. $$y_i=\beta+ 2 \beta x_i+\epsilon_i$$ $(...
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28 views

Expected Loss Calculation

How do you solve the integral $$E(L(\theta_A,\theta_B)) = \int_0^1\int_{\theta_B}^1(\theta_A - \theta_B)f(\theta_A)f(\theta_B)d\theta_Ad\theta_B$$ where $\theta_A \sim Beta(\alpha_1, \beta_1)$ and $\...
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Unit Step Functions

The image below shows the process for calculating the Expected value of a squared random variable X. From line 3 to 4 the unit step function [u(x) - u(x-1)] is removed and a 3 is placed before (1-x^2)....
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What's the best estimator for expectation if we can draw samples iid *and* we know the likelihood of each sample we receive

Suppose that $X_1, X_2, \ldots X_n$ is a sequence of random variables on a set $S$, drawn independently according to the pdf $p : S \to [0,1]$. Part I: Given some $f : S \to \mathbb{R}$, I want to ...
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Expected value of a conditional probability

I am reading this article: https://www.sciencedirect.com/science/article/pii/S0040580901915424?via%3Dihub Where as far as I can make out they are taking the expected value of the difference of 2 ...
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Example of a non-negative discrete distribution where the mean (or another moment) does not exist?

I was doing some work in scipy and a conversation came up w/a member of the core scipy group whether a non-negative discrete random variable can have a undefined moment. I think he is correct but do ...
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Expected value of Log Gamma function transformed Poisson random variable

This question came up during my research. In detail, let $X$ be Poisson distribution, i.e. $X\sim Pois(\lambda)$. I am interested in the expected value $$ \mathbb{E}\left[\log\left(\frac{\Gamma(X+c)}{...
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1answer
28 views

Geometric interpretation of mathematical expectation of a random variable

Is there any nice geometric interpretation of the mathematical expectation of a random variable (preferably based on density or cumulative density plot)? (For example, median has a nice geometric ...
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375 views

Are there any distributions other than Cauchy for which the arithmetic mean of a sample follows the same distribution?

If $X$ follows a Cauchy distribution then $Y = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ also follows exactly the same distribution as $X$; see this thread. Does this property have a name? Are there ...
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1answer
24 views

Fundamentals of applied statistics [duplicate]

I'm a begginer in statistics. I came across a problem, please help me out. Is expectation always mean arithmetic mean of a variable?
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1answer
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What's done with the expectations in this proof?

This is a proof of the per-decision importance sampling (theorem 1) from the appendix of: https://www.google.co.uk/url?sa=t&source=web&rct=j&url=http://scholarworks.umass.edu/cgi/...
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One-sided measurement error: $\widetilde{X} = X - \eta$, $\eta\geq0$. Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?

Let $X\geq0$, $\eta\geq0$ and $X,\eta$ independent. We measure $X$ with a one-sided error: $\widetilde{X} = X - \eta$. Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?
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On expected values of triple Kronecker Products

Consider a random vector $\boldsymbol{x} \in \mathbb{R}^N$ and the identity matrix $\boldsymbol{I}_N \in \mathbb{R}^{N\times N}$. I have to compute the expected value of the following Kronecker ...
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42 views

Expectation of Maximum Value

I'm trying to understand the basic statistics involved in trading. Suppose I'm trying to decide whether to buy a stock whose current price is $V_0$. Suppose I have some fancy statistical model from ...
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1answer
32 views

Need help with expected value calculation in coupon collectors in coupon collectors variant

I am revisiting a question I asked previously with a slight caveat. In my new situation, I am considering the marbles to always be attached to the same neighbors. Hopefully this will be clearer with a ...
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29 views

Expected value of counting function over infinite set

You have a random variable $X$, which can take a value from $i = \{1,2,3,...\}$, where the probabilities are $p_i = 0.5^i$. Now you if we have a list $(x_1, ..., x_n)$ of independent observations of ...
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Why is Expected value of a random variable equal to the mean

This is a derivation I found online for the mean of a random variable: Let $X_1, X_2, \dots, X_n$ be $n$ independently drawn observations from a distribution with mean $\mu.$ Let $\bar X$ be the ...
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Determining average life expectancy from a simple model

I'm trying to determine the average life expectancy from the following model. I feel I'm close, but I can't quite get the right answer. The daily death chance for any given individual is modeled as: ...
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Do discriminative models model conditional expectation?

In Machine Learning classic models like MLP, Logistic Regression or Linear Regression are called discriminative models. I frequently read that those models estimate the conditional probability ...
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48 views

How can I find the expected value of these dependent values?

Okay so there's a game, where you have 1 percent chance of winning an alpha pack and if you fail you your chance goes up by 3 percent. How would you calculate the expected value? I've only taken an AP ...
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Prove identity involving two multivariate normal distributions

Let $y_{ij}$, $i=1,2,\cdots,n_j$ be a random sample from $N_p(\mu_j,\Sigma_j)$, $j=1,2$. Let $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$ $$S_j=\frac{1}{n_j-1}\sum_{...
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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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Fourth order moments of the sum of multivariate normal distribution

Suppose $X\sim \mathcal{N}_d(\boldsymbol{0},\boldsymbol{\Sigma})$ follows a $d$-dimensional multivariate normal distribution. Let $\text{A}_i$ and $\text{A}_j$ be two arbitrary symmetric $d\times d$ ...
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Truncation and a composite distribution

Suppose $X$~$N(a,1)$ $Y|X$~$N(X,\sigma^2)$ Then what is $X|Y<0$ ~ ? and $E[X|Y<0]$ ~ ?
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Probability model EV

I'd like some help setting up a probability model and determining it's EV, but I don't really know where to start. Say there are 6 ranks, beginning at 0 and ending at 5. For each rank ...
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Related to Derivation of EPE in “The elements of statistical learning”

There is a question about EPE at StackExchange where next expresion is indicated: $$\mathbb{E}[(Y-f(X))^2]=\mathbb{E}[(Y-\mathbb{E}[Y|X])^2]+\mathbb{E}[(\mathbb{E}[Y|X]-f(X))^2]$$ I don't ...
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How to verify data in a linear fit for a sensor? Experimental vs Expected data

I am trying to characterize a sensor; in which I am trying to obtain the time constant as a function of the pressure applied to the sensor. I fitted a linear trend line into this data. However, I was ...
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1answer
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finding expected value when a set is given and a subset of size n is chosen

I found an interesting coding challenge on pramp by a friend but I couldn't do it in time. Anyhow, it says given a set { 3,14,7,22,29,33} and random 3 element subset is generated each time and its ...
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Is there any simple way for the expectation of squared of ratio of random variables?

So, for two random variables $A$ and $B$, I know that $\mathbf{E}[A]$, $\mathbf{E}(B)$, $Var(A)$ and $Var(B)$, I also know that $\mathbf{E}[A] = \mathbf{E}(B) + C$, I wonder if I can find a simple way ...
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Unbiased sampling and subsampling

Suppose I have a distribution $\mathbb{F}$ with mean $M$. Also, assume we have a set of i.i.d samples of size $n$ denoted by $X=\{x_1, x_2,..., x_n\}$ from $\mathbb{F}$. As a result, all $x_1, ..., ...
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$\mathbb{E}(\log(X_{max}/X_{min}) )$ of Weibull(alpha, 1)

I'm trying to find the expectation of log(max/min) from n samples of Weibull(alpha, 1). But I keep failing. Can anybody give some hints? I tried: $\mathbb{E}(\log(X_{max}/X_{min}) ) = \int_0^{\...
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1answer
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Expectation of latent variables in Factor analysis Model

I am going to through the theory behind factor analysis models given here Let's say our model is \begin{align} y_i = \mathcal \Lambda x_i +\epsilon, \end{align} where $y_i$ is the $p$-dimensional ...
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Finding a function minimizing the expected value

This is a question from an old Ph.D Qualifying Exam. Let $X,Y$ be two random variables. Find $g^{*}(X)$ so that $$\min_{g(x)}E(Y-g(X))^2=E(Y-g^{*}(X))^2$$ My attempt: If $g$ were a function of $Y$, ...
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Why temporal difference (TD) method has lower variance than Monte Carlo method?

This question might be a little trivial. However, I had a hard time understanding it or finding some formal proof for it. In many papers, it is being said that for estimating the value function, one ...
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1answer
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From expected value of R2 in Regression to expected value of partial eta squared in ANOVA

In general, $R^2$, the estimator of "multiple correlation coefficient" in regression, is known to be positively biased. Given $K$ predictors, and $N$ total sample size, Johnson, Kotz, & ...
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Expected probability with rigged sub probabilities

If you flip a fair coin 10 times, you could get TTTTTTTTTT. It is unlikely, 0.5^10, but possible. I've played video games that keep the 50% chance, but rearrange the odds so that you can never go more ...
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Equivalence between Expectations

Let $A$ and $B$ be a random variables with continues PDF $f_A$ e $f_B$. Let $Y=A+B$ and let $\hat{A}(Y)=Y$ be an estimation of $A$. I have to evaluate expectation of the square of the estimation ...
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Probability: $N$ dependent Bernoulli trials?

I understand the proof, but I don't see why Bernoulli trials are mutually independent [...] is not necessary.
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Expected distance of a stone thrown into a circular pond

A stone is thrown into a circular pond of radius 1 meter. Suppose the stone falls uniformly at random on the area of the pond. The expected distance of the stone from the center of the pond is: A) $1/...
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Is there any correlation between forecasting errors from forecasting done at different origins?

Let $\ e_{T+l|T} = Z_{T+l} - \hat{Z}_{T}(l)$ be the forecasting error $\ l$-steps ahead when the forecasting origin is time $\ T$. Now, let $\ e_{T+l-j|T} = Z_{T+l-j} - \hat{Z}_{T-j}(l)$ be the ...
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Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers $Y_1, Y_2, ...$ from the uniform distribution on $[0, 1]$, until ...
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Calculating conditional expectation and mean time to failure

I was reading text on probability where they state: $$\operatorname{Ex}[C]=\sum_{i=1}^{\infty}i\cdot\Pr[C=i]=\sum_{i=0}^{\infty}\Pr[C>i]$$ Now assuming there is a system which fails at each step ...
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How to find the expectation $\mathbb{E} \left[ \frac{|h|^4}{|h+w|^2} \right]$?

Given the independent and complex Gaussian random variables $h$ and $w$, how does one can find the following expectation? $$\mathbb{E} \left[ \frac{|h|^4}{|h+w|^2} \right] = \int_{\mathbb{C}}\...
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Brain-teaser: What is the expected length of an iid sequence that is monotonically increasing when drawn from a uniform [0,1] distribution?

This is an interview question for a quantitative analyst position, reported here. Suppose we are drawing from a uniform $[0,1]$ distribution and the draws are iid, what is the expected length of a ...
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Using a Dirichlet distribution to estimate an expected response and uncertainity of response

I have an $N$ dimentional Dirichlet distribution $\text{Dir}\left(\alpha\right)$ that describes the number of times different responses, $r\in\left(1, 2, 3, \ldots N\right)=R$, have been observed. I ...