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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45 views

Shrinkage of the Sample Covariance matrix

Assume we have N independent and identically distributed random vectors $X_1, X_2, ..., X_N$ where each of them is of size p $\times$ 1. The sample covariance matrix, denoted here by $S$, is computed ...
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2answers
54 views

How to advise using probabilty [on hold]

A game of chance entails three coins. For each head that comes up, the player receives £10 and loses £8 for any tail that comes up. To play the game, one pays £15. Advise a player whether it is ...
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14 views

conditional/unconditional expectation and variance for an AR(1) process

We have an AR(1) process, $X_t=\alpha X_{t-1}+\varepsilon_t$ with $\varepsilon\sim(0,\sigma^2)$, $X_0=0$ and $|\phi|<1$. We have the conditional expected a value with respect to $X_{t-1}$: ...
7
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56 views

Expected value of product of non independent Bernoulli random variables (correlations are known)

I've asked a question about getting the joint probability distribution for $N$ Bernoulli random variables, given the expected value for each one ($E[X_i]=p_i)$ and it's correlations ...
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2answers
40 views

Working with expected value of a matrix

If I have two matrices $C$ and $D$ of the same size. If I know that the expected value of $C$, denoted by $E(C)$, is equal to $D$. So $E(C)=D$. In this case, $E(diag(C))$ will be equal to $diag(D)$, ...
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4 views

Find the expected frequency of some state in a state sequence of length N given a transition matrix M

I can represent stochastically-articulated sequences of states using a transition matrix M where a given entry in cell (i,j) corresponds to the probability of state j given that the current (or, most ...
5
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2answers
115 views

What is the expected value of $\frac{X}{X+Y}$?

I am trying to find the expected value of $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg]$. I started with writing $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg] = ...
4
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1answer
100 views

Expected value of tangent of a normal random variable

If $z\sim N(\mu,\sigma^2)$ What is $E[\tan(z)]$ and $E[\tan^2(z)]$? Generally, it seems that the expectation does not exist. How about if $z$ is bounded $(0,\pi/2)$? Update: Theoretically, my ...
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45 views

CLT and 2 variables

Okay so there are 2 variables $D_i$ and $V_i$. Now $D= D_1 + D_2 + ... + D_N$ and $V = V_1 +.. +V_N$ Now I know the relationship is such that $E[D_i - a*V_i] = 0$ and $Var[D_i - a*V_i] = E[D_i]$ ...
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23 views

What is expected count formula for zero-inflated negative binomial regression?

My IT department wants me to translate my zero-inflated negative binomial regression model into a formula for calculating expected count which they can hard code into SQL. I'm running the model in ...
0
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1answer
31 views

Given a set of numbers from 0 to 100, what's the chance that the next number will be less than 10?

I performed a set of trials. Every trial returned a number 0<=N<100. What's the chance that in the next trial, the picked number will be less than 10?
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40 views

When do Taylor series approximations to expectations of (entire) functions converge?

Take an expectation of the form $E(f(X))$ for some univariate random variable $X$ and an entire function $f(\cdot)$ (i.e., the interval of convergence is the whole real line) I have a moment ...
0
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1answer
51 views

Getting Expected Value from Monte Carlo Simulation

There are two independent uniform continuous random variables $X$ and $Y$ (such that $0 \leq X \leq 10$, $0 \leq Y \leq 10$). The function $f$ is the difference between the two random variables ...
0
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1answer
25 views

Double expected value, which comes first?

In the following equation, the outer expectation is over the distribution $X_i|T_i = 1$ $\tau|_{T = 1} = E(E(Y_i|X_i, T_i = 1) - E(Y_i|X_i, T_i = 0)|T_i =1)$ Are we taking the expected value of ...
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0answers
5 views

How do I find the expected coverage of the domain after K ranges are selected?

If I have a domain of 0 to 1000000 (but I doubt the domain matters, just that the ranges can overlap). I then chose 1% ranges uniformly, allowing for ranges to overlap completely or partially. How do ...
2
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1answer
46 views

How to estimate a chance of getting in the first positions, given previous tries?

I perform a series of experiments that return a number N that is between 1 and MAX. MAX varies between experiments. For example: ...
3
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1answer
51 views

$E_{\theta_1}[\ell(\theta_2;X)] $

I am faced with the following in my "Statistical inference book" $E_{\theta_1}[\ell(\theta_2;X)] $ where $\ell(\theta_2;X)$ (loglikelihood) is $\log [P(\theta_2;X)]$, X is a random variable. What ...
2
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1answer
47 views

Expected value of a diagonal

If I have $E[A] = B$, where $E$ is the expected value, $A$ and $B$ are square matrices and $\text{diag}$ refers to the vector of coefficients on its diagonal. In this case, what will be the value of ...
2
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0answers
24 views

Non significant A/B test, what is the future expected value?

Suppose you run an A/B test with a result that is not significant. For example: Version A: 10,000 views, 100 clicks Version B: 10,000 views, 115 clicks What is ...
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0answers
24 views

Expectation over training and testing set

If I have a linear regression model with $p$ parameters that fit by least squares a training set $(x_1, y_1),...,(x_n, y_n)$ drawn at random from a population, and we have some test data ...
2
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1answer
65 views

Computation of expected values

I have a $N$-dimensional (normal) random vector $\mathbf{X}$, where $N$ is large, and a function $f : \mathbb{R}^N \to \mathbb{R}$. My goal is to compute $\operatorname{E}[f(X)]$ or at least ...
6
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1answer
114 views

Expected number of times you spent in a state of an absorbing markov chain, given the eventual absorbing state

It's well known that, if $Q$ is the matrix of transient state transition probabilities, and $$ N = \sum_{n=0}^{\infty} Q^n = (I - Q)^{-1}$$ then $N_{ij}$ describes the expected number of times the ...
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1answer
32 views

How do i show multiplication of covariances?

i am trying to show the following $$Cov(a,b) = \frac {Cov(a,x) * Cov(b,x)} {Var(x)}$$ I am a bit lost on how to expand the numerator term, so i wrote the following $$Cov(a,b) = \frac ...
0
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1answer
31 views

Time to completion

Say a person needs to go from A to B and also needs to complete a task. He may complete the task at either A or B. The task, say, takes 10 minutes. It takes 60 minutes to go from A to B after boarding ...
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3answers
145 views

Expectation of $b^T \operatorname{sign}(Ab)$

I'm trying to compute the expectation of: $$b^T \operatorname{sign}(Ab)$$ Where $b$ is a $n\times1$ vector of independent Bernoulli random variables: $$P(b_i = 1) = 0.5,\quad P(b_i = -1) = 0.5$$ and ...
2
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1answer
44 views

Shannon entropy and inequality of expectations

Consider two distinct probability distributions $P(X)$ and $Q(Y)$---defined on the same domain---with (Shannon) entropy of $H(X)$ and $H(Y)$. I am interested to prove that $$ H(X) \leq H(Y) \implies ...
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48 views

Using Dirac Delta functions for estimating a probability distribution

I'm having some trouble understanding this slide. It's mentioned in the context of gaussian distributions. I sort of understand the Dirac delta "function". The main difficulty I'm having is with ...
0
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1answer
47 views

How to compute $P(|X - E_Y[h(y)]| < c)$?

Consider a discrete random variable $Y$, a continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, and ...
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29 views

bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \stackrel{iid}{\sim} P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one ...
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31 views

Relationship between expectation and probability with continuous variables

I've tried to word this question to the best of my ability but I may have got some of the terminology wrong. Example: I have a machine producing various lengths of knotted string. If I have a string ...
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2answers
134 views

Is it true that $E[Y|W]=E[Y|E(X|W)]$, given that $W$ is $X$ measured with error

In Carroll's 2006 book "Measurement Error in Nonlinear Models", and on p38 it is stated without proof that: [One can] estimate the regression of $Y$ on $(Z, X)$ and then to substitute into this ...
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1answer
34 views

Expected number of draws from an urn (without replacement) to get 2 different coloured balls

An urn has $N=18$ black balls and $M=2$ white balls. What is the mean number of balls one must draw to get two different colours. My solution so far is extremely ugly (though effective): ...
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57 views

What is the theoretical difference between using population and sample means for calculating expectation?

In looking at linear combinations of random samples, what is the bridge between substituting the population mean $\mu$ with the sample mean? For instance: Population mean (sample size n): $$E(Y) = ...
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1answer
31 views

What is the meaning of averaging combinations of independent random variables?

I'm getting into the management of multiple independent random variables in determining expectation and variance, but cannot see where averaging of a linear combination of independent random variables ...
4
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3answers
144 views

In which case $\mathbb E[X]=\sum _ix_i P[x_i]$ can be $0$ when all $x$'s are not zero ($0$)?

Say $X$ is a random variable and $x$'s are realizations of $X$ . Say , $\mathbb E[X]=\sum _ix_i P[x_i]=0$ . But I do not understand in which case $\mathbb E[X]=\sum _ix_i P[x_i]$ can be $0$ when all ...
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4answers
167 views

Why is $\mathbb E(X)=\sum_{i=1}^{n}x_i P(x_i)$?

If $X$ is a random variable and $x$'s are the realizations form $X$ and $N$ is the population size $n$ is the sample size Which one is correct $\mathbb E(X)=\sum_{i=1}^{N}x_i P(x_i)$ or ...
2
votes
0answers
39 views

Why moments of expectation are known as “moments” [duplicate]

I am studying moments of expectation, and seen the formulas for computing the moments. There is one thing I am not clear of, and not getting answer for that. Why moments are named as moments? To my ...
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1answer
55 views

Example Bayesian resolution of the Two Envelopes Problem [closed]

What is a concrete example of a Bayesian resolution to the Two Envelopes Problem?
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29 views

A step of a proof regarding the Nadaraya-Watson estimator

Let the data be $(y_i , X_i) $ where $y_i$ is real valued and $X_i$ is a q-vector. The regression function for $y_i$ on $X_i$ is $g(x) = E(y_i | X_i = x)$, we can write this as: $$y_i = g(X_i) + ...
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vote
1answer
19 views

Inequalities for maxima over random functions

Let R be the set of real numbers. Say we have a function $f(x,X) \in R $ where $X \in R^d$ is a random variable over $\Omega$ and $x \in R$. I'm searching for an upper bound for the expected value of ...
2
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1answer
45 views

How can one resolve an apparent paradox regarding the uncertainty of the product of two measured quantities?

Suppose one has three quantities $X$, $X_1$ and $X_2$, such that $X = X_1X_2$. Since percentages uncertainties of products just add up we have: $$\frac{\delta X}{X} = \frac{\delta X_1}{X_1} + ...
4
votes
1answer
118 views

What is the expectation: E[(2X + 3)^2 ], given E[X] = 1?

I'm taking an upper level Economics class and one of my assignments asks the question in the title. I approached it by using one property of expectation: expectation of the sum is equal to ...
5
votes
3answers
381 views

Expected number of dice Rolls require to make a sum greater than or equal to K?

A 6 sided dice is rolled iteratively. What is the expected number of rolls required to make a sum greater than or equal to K? Before Edit ...
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0answers
58 views

Expected value of a function of an exponential random variable

I want to find the expected value of a non-decreasing cost function $c$ of an exponentially distributed random variable $X$ with mean $a$. Considering some constant $t$, the expected value that I ...
9
votes
1answer
149 views

What is $\mathbb E \lVert X \rVert$ for a multivariate normal $X \sim \mathcal N(\mu, \Sigma)$?

$\DeclareMathOperator\E{\mathbb E} \DeclareMathOperator\Var{\mathrm{Var}} \newcommand\R{\mathbb R} \DeclareMathOperator\N{\mathcal N} \DeclareMathOperator\tr{\mathrm{tr}}$Suppose $X \sim \N(\mu, ...
5
votes
1answer
222 views

Degeneracy paradox

Say I have a highly biased coin that lands heads with $p_h=0.01$ and tails with $p_t=0.99$, and I flip it $98$ times. The probability of zero heads is ${p_t}^{98} \approx 0.373$. The probability of ...
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1answer
43 views

Expectation with transformed random variable

The expected value of a function $f$ of a random variable $x\in R^n$ is defined as $E_x[f(x)]=\int p(x)f(x)dx$ where $p()$ is the pdf of $x$. Assume a function $h: R^m \rightarrow R$. Let $z=Mx$, ...
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0answers
54 views

How to calculate probability of bankruptcy for $E>0$ and given $\sigma$?

I will demonstrate the problem with the simple example. Suppose I want to start a casino. The probability theory tells me that the expected value for my games is greater than zero ($E>0$), and ...
0
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0answers
22 views

Help please regarding covariance

I have a question regarding covariance. If we have two independent variables $X$ and $Y$, then is the $Cov(X^n, Y^m) = 0$ with arbitrary values of $n$ and $m$? I know that $Cov(X^n,Y^m) = E(X^nY^m) ...
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20 views

Finding the expectaion with respect to a gaussian measure on “half” of $R^n$

I want to calculate the following expectation: $$ \int_{X_\theta} x\phi(x) dx $$ where $\phi$ is the density of a (not necessarily standard) gaussian distribution and $$ X_\theta = \{ x \in ...