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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$\langle x_1 \rangle > \langle x_2 \rangle$ implies that $\langle f(x_1) \rangle > \langle f(x_2) \rangle$ ? if $f(x)$ is an increasing function

Let $f(x)$ be an increasing function. Consider two random variables $X_1$ and $X_2$, with different probability distributions, such that $\langle x_1 \rangle > \langle x_2 \rangle$. Does it follow ...
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Definition of mutual information

The mutual information between two random variables X and Y is given by $I(X,Y) = E\bigg [ \ln\bigg ( \frac{p(x,y)}{p(x)p(y)} \bigg) \bigg]=\sum_{x,y}p(x,y)[\ln p(x,y)-\ln p(x)p(y)]$ I'm trying to ...
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1answer
16 views

Expectation of a variable inside the cumulative distribution function of standard normal

Let $\Phi$ and $\phi$ respectively be the cumulative distribution function and probability density function of a standard normal distribution. $\beta$ is a $d \times 1$ vector which follows a ...
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0answers
10 views

How to properly project a value representing 38% to 100% to show potential range of true value? [on hold]

I have a value that represent 38% of user market data, I want to expand the 38% to 100% but doing so may very well be inaccurate because the 38% user behavior may be different than the rest of users. ...
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153 views

Why does the number of continuous uniform variables on (0,1) needed for their sum to exceed one have mean $e$?

Let us sum a stream of random variables, $X_i \overset{iid}\sim \mathcal{U}(0,1)$; let $Y$ be the number of terms we need for the total to exceed one, i.e. $Y$ is the smallest number such that $$X_1 ...
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34 views

Expected value of normal-wishart, is the solution correct?

I want to compute expected value of $E[μΛ]$ for a normal-wishart distribution how can i compute it? A normal-wishart distribution is defined as below: $$ NW(μ,Λ|μ_0,λ,W,v)=N(μ|μ_0,(λΛ)^{−1})W(Λ|U,v) ...
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Dummy coding: code two values $a$ & $b$ instead of $0$ & $1$?

I know there are a lot of questions and answers related to dummy coding. But I still wonder if it matters to code a dichotomous variable d in this regression model $$y = \beta_1 + \beta_2 \cdot d + ...
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17 views

Expectation of dependent variables in normal-Wishart distribution

I want to compute $E[\mu\Lambda]$ for a normal-Wishart distribution how can I compute it? A normal-wishart distribution is defined as below: ...
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0answers
42 views

Write $\mathop {\mathbb E}[(X AX^h)]$ in function of $\mathop {\mathbb E}[(X X^h)]$?

Suppose $X$ is an $i \times j$ random matrix. In addition, $X$ has complex i.i.d. normal entries with $0$ as mean. We define $A$ (of dimension $j \times j$) as a deterministic matrix. Is it ...
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62 views

How to calculate $E(x)$ and $V(x)$ when $g(x)\sim f(g(x))$

Let assume that we are interested in a variable $x$. We know that e.g. $g(x)=x^2$, $g(x) \sim Uniform(a,b)$ or any other distribution. From that I can calculate $E(g(x)) = (a+b)/2$ and ...
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0answers
26 views

Relation between expectations of two random variables

I have two random variable X and Y. I know that $$E_X\left[X \log \frac{X}{e}\right] < E_Y\left[Y \log \frac{Y}{e}\right]$$ Using the above relation can I say anything about the relation between ...
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2answers
297 views

Expected value of sum of cards

If each card on a regular 52 deck card has points that corresponds to their number (like 2 of hearts is 2 points, 7 of clubs is 7 points), the Jack, Queen, King each being 10 points and you keep ...
2
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0answers
35 views

Expectation of $|U|$ given the first and second moments of $U$

Let U be a random variable with $E(U)=0,Var(U)<\infty$, is it true that $E(|U|)<\infty$?
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59 views

Random permutations [closed]

Let $a_1 < a_2 < · · · < a_n$ denote a set of n numbers, and consider any permutation of these numbers. We say that there is an inversion of $a_i$ and $a_j$ in the permutation if i < j ...
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1answer
42 views

Finding the expected value of the cdf?

I have this question on a homework assignment, and I'm not sure how to solve it. Assume $X \sim \exp(\lambda=2)$, define $Y=F(x)$, where F(x) is the cdf function of $X$. Calculate the expected value ...
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20 views

How to compute a mean of some numbers from normal distribution? (using continuity correction) [closed]

In practice, specifically writing a paper, we often need to compute an expected value of some sample drawn from a certain continuous distribution. Let's say, draw 100 numbers from $N(0,1)$, and ...
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83 views

Example of computing the expectation of a discrete RV using Riemann-Stieltjes integral?

Riemann-Stieltjes integral notation is used in expectation expressions in some probability texts. Basically, dF(x) pops up in the integral rather than f(x)dx in the integral, since the CDF F(x) may ...
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1answer
16 views

Calculating the expected number of unbalanced dice needed to reach a sum

I was recently commenting on game mechanics, when I realized that I actually underestimated this particular part. Given an unbalanced die (in this case, some of the sides are 0), what is the expect ...
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18 views

Proof for Irreducible Error statement in ISLR page 19

This section of Introduction to Statistical Learning in R (page 19 in v6, statement 2.3) is motivating the difference between reducible and irreducible error (that is noted by $\epsilon$ and has mean ...
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Compound Distributions — Basic Techniques and Key General Results from First Principles

Could someone please point me to a source with notation, terminology, key results and basic techniques to approach compound distributions? Definition Compound probability distribution is the ...
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33 views

Finding Probability Distribution Parameters Satisfying a Particular Condition [closed]

Condition: $$E[\text{max}(X,Y)] \leq E[\text{max}(K,Y)]$$ Here, $X,Y$ are random variables. $K$ is a constant. The distribution for $Y$ is known. Question one: Is it possible to find the ...
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1answer
49 views

Can't understand probability calculation appeared in “Mining of Massive Dataset” book

This is a summary of the excerpt: Given that we have: *$10^9$ people *each person has a chance of 0.01 of going to a hotel *we have $10^5$ hotels, each hotel holds 100 people *Thus, $P(2 \text{ ...
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112 views

Best Linear Prediction and variance decomposition

Let $( X , Y )$ have moments of at least the second order ,and let $\hat{Y} = a +bX$ then we will choose the coefficients $a$ and $b$ such that $a=E(Y)-\sigma_{XY}/\sigma_{X}^2E(X)$ and ...
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21 views

Asymptotic normality for nonsmooth objective functions

Assume that $f ({\bf x}; \theta): \mathbb{R}^p \times \Theta \to \mathbb{R}$, where ${\bf x}$ is the vector of inputs (with some distribution) and $\theta$ is the vector of parameters. Also, assume ...
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1answer
27 views

In which cases we can approximate expected value of a function by assuming the function and the expectation commute?

I was reading a computer vision paper and the authors approximated $\left<f(X)\right>_{Q}$ with $f( \left< X \right>_Q)$ where $f(\cdot)$ is nonlinear. Are there any rules of thumb for ...
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36 views

Find $\mathbb{E}\{M W V V^H W^H M^H \}$ in the following case

Let $M$ an $n \times p$ matrix with complex Gaussian elements with mean $= \mu$ and variance $= \sigma^2$. Also le define matrix $W$ as an $p \times m$ matrix for which the columns are of unit norm. ...
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2answers
63 views

E(x) for uniform distribution

I would like to take a uniform distribution for example. Let’s say a train will arrive at the station randomly within every 10 minute window, so the probability density function is $f(t)=0.1$,$\: t ...
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27 views

Generalized exponential distribution, an expectation

When i read an article about Generalized exponential distribution, i came across an expectation and i couldn't figure out how the authors obtain that result. The pdf is \begin{equation} ...
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29 views

With given numbers $a_1, a_2,a_3,…,a_N$, let $W=\Sigma_{i \in s_n}a_i$. Calculate the mean and variance of $W$

From the set $R=\{1,2,3,...,N \}$, a set $s_n$ of $n$ numbers are chosen without replacement, $0<n<N$. With given numbers $a_1, a_2,a_3,...,a_N$, let $W=\Sigma_{i \in s_n}a_i$. Calculate the ...
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26 views

Related to Chebychef's inequality

Please help me with this problem. Suppose that $X$ is a random variable for which $E(X)=\mu$. Prove that $$\Bbb{P}(|X-\mu|\ge t)\le \frac{E[(X-\mu)^4]}{t^4}$$ The only thing I have been able to do is ...
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140 views

Train waiting time in probability

Let's say a train arrives at a stop every 15 or 45 minutes with equal probability (1/2). What is the expected waiting time of a passenger for the next train if this passenger arrives at the stop at ...
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20 views

Building compound distribution of Bernoulli and geometric distribution

Let $ X \sim \text{Bernoulli} (N, p)$ where $N \sim \text{Geometric }(\theta)$ Then how to calculate P(X=x), E(X) and V(X)? If $N \sim \text{Geometric }(\theta)$ then $f(n) = \theta ...
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Sufficient statistic for expectation of exponential family

True or false: Let X $\sim$ f, where f is element of an exponential family. Then, $\frac{\sum_{i=1}^n x_i}n$ is a sufficient statistic for $E(X)$. For either case, please provide the ...
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1answer
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Computing the expected value for a sample

Assume that we have a dataset $D$ having $N$ instances: $X_1,\cdots,X_N$. We select $n$ items $(Y_1,\cdots,Y_n)$ using sampling without replacement. We use $\bar{Y}$ as an estimate for $\bar{X}$, and ...
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1answer
99 views

How to compute expectation of square of Riemann integral of a random variable?

How does one compute $E[(\int_0^T W_s ds)^2]$ where $(W_t)_{t \in [0,T]}$ is standard Brownian motion in $(\Omega, \mathscr F, \mathbb P)$? Apparently proving $$\int_0^T W_s ds = \int_0^T (T-s) dW_s ...
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1answer
60 views

KL divergence and expectations

I am trying to understand the explanation of the KL divergence per below. It refers, as i understand it, to an expectation in the second term. "Approximating the expectation over q in this term". ...
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1answer
44 views

Correlations between 4 variables

I have $m_{1R}$, $m_{2R}$, $m_{1L}$, $m_{2L}$, and about them, I know that $Corr(m_{1R}, m_{1L}) = 0$ $Corr(m_{2R}, m_{2L}) = 0$ $Corr(m_{1R}, m_{2L}) = 0$ $Corr(m_{2R}, m_{1L}) = 0$ Suppose I ...
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0answers
12 views

Trouble understanding equations with a mix of variances, expected values and means

Let's assume we don't know the real value $\mu$ of the average of a of random variable $x$. We can find an estimator for the variance using the Bessel correction: $$\widehat{V(x)} = ...
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19 views

Adjust Expected Value from Odds-Ratio

Suppose Group1 makes up 40% of a population, and Group2 the other 60%. Also suppose that the odds ratio for selection of Group1 compared to Group2 is .25. What would then be the expected number of ...
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59 views

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

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 $p>2$. What if instead we wanted to find the distribution of ...
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1answer
29 views

Population vs sample

When we think of linear regression, the implicit assumption is that we only observe a small fraction of a possibly infinite large population. Thinking of simple averages, imagine a fair die. The ...
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1answer
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Help to understand this. Expected value of $S^\alpha$ in Gaussian distribution

Lets $X_1,\cdots,X_n$ be simple random sample from $\mathcal{N}(\mu,\sigma)$. $\overline{x}$ is sample mean. Let $$S^2=\begin{cases}\sum_{i=1}^n (x_i-\mu)^2, \mathrm{ where\ } \mu \mathrm{\ is\ ...
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1answer
99 views

Expected value of iid random variables

I came across this derivation which I don't understand: If $X_1, X_2, ..., X_n$ are random samples of size n taken from a population of mean $\mu$ and variance $\sigma^2$, then $\bar{X} = (X_1 + X_2 ...
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45 views

Expected value of a censored poisson

I have the following density (it's a censored model): $$f(y) = \begin{cases} e^{-\lambda}(1+\lambda), & \text{if }y^*=0,1 \\ \frac{\lambda^{y^*}e^{-\lambda}}{y^*!}, & \text{if }y^*=2,3.... ...
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19 views

Estimate values from a multivariatie normal distribution

I have a multivariate normal distribution with parameters mean = 0 and a variance-covariance matrix K which is n-by-n. I'd like to generate a vector of values (length n) that represent the expected ...
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28 views

Expected value of a marginal distribution when the joint distribution is given

I am asked to find the expected value of a vector of two random variables when the joint density is given. Is the recipe for solving this problem: Find the marginal distributions Find the expected ...
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1answer
78 views

Expected value of $e^{X}$

I am trying to find the expected value of $Y=e^{X}$ where the density of $X$ is $f(x) = 2x$ for $0<x<1$ (zero elsewhere). According to my textbook, the answer should be $2$. I get the correct ...
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1answer
76 views

Getting expected value from Standard Deviation and Skew?

If I have: - Standard deviation - Skew What additional information can I calculate ? Could I calculate Expected Value?
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19 views

Apply gradient descent on expected valued cost function

I want to use gradient descent algorithm to minimize the Kurtosis of error function. So, I need to calculate the gradient of cost function $J$. Below $\boldsymbol{w}$ and $\boldsymbol{x}$ are same ...
3
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3answers
54 views

How do I find the expected value of the sum of a function?

So for example, if I had to find $E(\sum_1^n\log(X_i))$ would this be equal to $\sum_1^nE(\log(X_i))$ and then do I proceed from there? By the way, $X_1, X_2, ...,X_n $ is a random sample from the ...