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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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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29 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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1answer
89 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
30 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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32 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
28 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 ...
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3answers
138 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
160 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 ...
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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
47 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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17 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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1answer
16 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 ...
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1answer
42 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} + ...
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1answer
106 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 ...
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3answers
338 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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38 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 ...
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1answer
139 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, ...
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1answer
217 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
38 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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41 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 ...
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21 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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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 ...
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1answer
42 views

Expected value or median of a function defined on sphere with uniform distribution

Suppose $S$ is the sphere of radius one. And suppose $f:S\to \mathbb{R}$ is defined as follows: $$f(x_1,...,x_n)=\frac{1}{x_1^2}+\frac{1}{x_2^2}+\cdots+\frac{1}{x_n^2}$$ I am trying to calculate ...
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98 views

$E(X)E(1/X) \leq (a + b)^2 / 4ab$

I've worked on the following problem and have a solution (included below), but I would like to know if there are any other solutions to this problem, especially more elegant solutions that apply well ...
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33 views

Expected value of $e^{\alpha \sqrt{t-s} \ Z}$

How can I find the expected value of this: $e^{\alpha \sqrt{t-s} \ Z}$, where Z is a standard normal random variable. I know the moment generating function should help me with this, but I can't ...
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1answer
37 views

How does this expected value translate into a conditional variance?

I'm working with a simple local level model in a textbook \begin{align} y_t &= \alpha_t + \epsilon_t, \qquad \epsilon_t \sim N(0, \sigma_\epsilon^2) \\ \alpha_{t+1} &= \alpha_t + \eta_t, ...
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1answer
53 views

What is the relationship between the mean of a continuous variable and expected value?

What is the relationship between the mean of a continuous variable and expected value? How are they connected?
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21 views

Transformation Theorem & Piecewise Function

I am a total statistics newbie and I hope that you can help me with the following problem: Let $X \sim \mathcal N\left(\mu, \sigma^2\right)$ be a random variable. Define a new random variable $Y$ as: ...
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1answer
37 views

Finding expected order statistics from a normal with known parameters [duplicate]

I'm interested in finding the expected value for the kth ordered observation of a normally distributed variable with known standard deviation, mean and n. Could someone let me know the formula for ...
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1answer
37 views

Chi-squared distribution and dependence

We know that for a group of independent random variables $\{X_i\}_{i=1}^n$ s.t each $X_i$ is distributed as a chi-squared distribution with degree of freedom one ($\chi_1^2$), the sum of these random ...
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Joint Gaussian Distribution Question

Drawing a pair of $(x, y)$ from a joint Gaussian distribution with $r$ covariance. Knowing the standard deviations of $x$ and $y$ and knowing $z = x + y$, what is your best guess for $x$?
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1answer
117 views

An 'easy' exercise on conditional expectations and filtrations

I am struggling with the following exercise in the context of modeling information structure via filtration to evaluate contingent claims. I hope that someone can explain me how to derive the ...
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1answer
53 views

Upper Bound on $E[\frac{1}{1-X}]$ where $E[X]=a$ and $0<a<1$

$X$ is a discrete random variable that can take values from $(0,1)$. Since $\varphi(x)=1/x$ is a convex function, we can use Jensen's inequality to derive a lower bound: $$ ...
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1answer
83 views

Expected value of Y = (1/X) where $X \sim Gamma$

I'm having some confusion over this statement here. Let $T_i \sim Exp(\lambda + \theta)$ and if they are all iid then $\sum_n T_i \sim Gamma(\alpha = n, \beta = 1/(\lambda + \theta))$ I want to find ...
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1answer
45 views

Exponential distribution of the time between recurrences

Patients in a hospital are treated for certain illness. The time in days that no recurrence takes place follows exponential distribution with mean of $\theta = 27$ days. The following table gives the ...
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computing expectations in variational updates

I have a complete log-likelihood expression as follows: $$ L = \sum_{i=1}^N \log P(y_i|x_i, w_i, \beta) + \log P(\beta) + \sum_{i=1}^N \log P(w_i) $$ Now, I need to compute the expectation of these ...
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difference between sampling stopping times

Let $\{a_n\}$ be an increasing sequence of nonnegative numbers. Let $Z_i$ be a sequence of iid rvs. Define $T_1=\inf\{n\ge1: \sum_{i=1}^n Z_i\ge a_n\}$ and $T_2=\inf\{\text{ODD}\,\,n\ge1: \sum_{i=1}^n ...
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Expectation maxmisation algorithm increases true likelihood at each iteration

I've heard that the EM algorithm ensures that the true likelihood is non-decreasing at each iteration of the algorithm, but I'm not sure why this is the case. I've provided a basic plot which I ...
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1answer
42 views

Simple question about finding conditional expectation

Let $X,Y \sim U[0,1]$ ($X,Y$ are independent), we want to find $E[X|X>Y].$ I tried a few approaches to the above problem, but am not confident in my answer. One approach is as follows. Note that ...
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31 views

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, ...
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Expectation of cube of summation of independent random variables

Where would I begin on this problem? I know I begin with pulling $c^3$. Where would I go from there? And I know that $\mathbb{E}[X] = x_1p_1 + .... x_n p_n$ I'm stuck on the rest, however.
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Deriving the probability function from a logistic regression model

First off, I don't know a lot of stats. That said, I am hoping you can help me derive the function to calculate the fitted probability from the summary of a logistic regression below: My instinct ...
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1answer
57 views

Continuous random variable and median problem

The definition of median, $m$, for a continuous random variable $X$ is $$P(X\leq m)=P(X\geq m)=\int_{-\infty}^m f(x)\text{d}x=\int_{m}^{\infty}f(x)\text{d}x=\frac{1}{2}$$ where $f(x)$ is the pdf of ...
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Expectation of output of an LTI system w.r.t. a WSS random process

Let $X(t)$ be a wide-sense stationary random process―i.e., its expectation is a constant and its autocorrelaton function is a function only of time differences―and let $Y(t) = X(t) * h(t)$ where ...
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1answer
68 views

Finding expected value

I am not sure of how to find the value asked in below question. Any help would be appreciated. Suppose that the joint distribution of $X$ and $Y$ is the uniform distribution on the circle disc ...
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272 views

Expected number of tosses till first head comes up

Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time. What is the expected number of tosses that will be required? What is the expected number of tails that will ...
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55 views

Is there a formula for a general form of the coupon collector problem?

I stumbled across the coupon collectors problem and was trying to work out a formula for a generalization. If there are N distinct objects and you want to collect at least k copies of each of any m ...
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94 views

Show that $E[\mu_i^2|X_i]=\sigma^2$ and $Var(\mu_i|X_i)=\sigma^2$

I'm a little stuck on this review problem so help would be greatly appreciated! Q: We have the regression model $Y_i=\beta_0+\beta_1X_i+\mu_i$ and we assume that the expected errors are $0$. We also ...
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1answer
57 views

How to solve this Expectation of log of random variable

This may seem a trivial Question but I am confused and never come across this kind of expression where I need to simplify for a function of a random variable $R$. I have an expression $E\bigg ...
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20 views

Learning Expected value

We probably have played the game "Throwing Balls into the Basket". It is a simple game. We have to throw a ball into a basket from a certain distance. One day we were playing the game. But it was ...