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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71 views

Is it okay to write the square of expectation of a random variable $X$ as $\mathbb{E}^2(X)$?

Is this notation accepted when I write $\text{Var}(X)=\mathbb{E}(X^2)-\mathbb{E}^2(X)$?
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11 views

Expected value of a semi-partial correlation

Say I have 4 random variables. $X^{(1)}$ and $X^{(2)}$ are jointly multivariate normal with mean 0 and covariance $\Sigma_X$, and $Y^{(1)}$ and $Y^{(2)}$ are jointly multivariate normal with mean 0 ...
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20 views

Handling the “positive bias” in percentages of positive variables

Say we run an experiment and notice the following impact on a variable of interest (one row per experimental unit): ...
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14 views

Expectation of infinite sum of random wishart matrices

I have problem figuring out how to find the expectation for an infinite sum of random matrices. More explicitly, my problem is: Let $\mathbf{S}_i$ be the maximum likelihood estimator of the sample ...
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1answer
34 views

Expected Value clarifications using R/Excel

So it has been a little while since I've taken my last statistics course, and I wanted to double check that I am not making any kind of grave errors in my expected value calculations. Quick bit of ...
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30 views

Expected number of points in k turns?

I'm not too familiar with expected values so I'm wondering if people could verify if I'm on the right track with my thought process here. If the probability of winning one point in a turn is 1/9, ...
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33 views

Minimising MSE of $\sigma^2$ estimator of specific form

I have found a past exam question for a statistics course and can't seem to find the required result. Part A is fine but my working for part B must be incorrect [see below]. Can anyone figure out ...
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1answer
43 views

How would you calculate $E[\mid x \mid ^{\alpha }], \alpha \in \Re$?

Here $x \sim N(0,1)$. I realize that the expectation won't be defined for $\alpha$ when the integral goes to infinity. I can't seem to figure out which specific values of $\alpha$ would cause this. ...
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49 views

Finding the expected value of a continuous random varibale when the commulative distribution is given

I have this distribution function of a random variable X: I wish to find E(X). I have used derivatives to get the density function, compared it to 1, and found that f(t) = (4/5)t+(3/5). I then ...
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36 views

Conditional expectation of a univariate Gaussian

Suppose I have a univariate Gaussian distribution with mean $\mu_X$ and standard deviation $\sigma_X$, and I know the random variable $X$ is least some positive value $y$: $X \geq y$. What is the ...
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72 views

How to find $E|X+Y|^3$ from related information?

Assume that $$ E(X+Y)=E(X-Y)=0 $$ $$ V(X+Y)=3 $$ $$ V(X-Y)=1 $$ Show that $E|X+Y|\leq\sqrt3$. If in addition, it is given that $(X,Y)$ is bivariate normal, calculate $E|X+Y|^3$. For the 1st part, ...
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63 views

Expectation with indicator function

I have the following expectation $$E[x_{t+1} \mathbf{1}_{\{x_{t+1}> z_t\}}]$$ where $x_{t+1}$ is a normally distributed random variable $x_{t+1}\sim N(0,\sigma^2)$, and $\mathbf{1}$ stands for ...
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92 views

Expected value of maximum ratio of n iid normal variables

Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma^2)$ and let $X_{(i)}$ denote the $i$'th smallest element from $X_1,...,X_n$. How would one be able to upper bound the expected maximum of the ratio ...
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33 views

Expectation of a hazard rate

I need to estimate the expectation of a hazard function, $E[h(x)]$. For instance, for the exponential distribution the result is equal $\lambda$ $E[h(x)]=\int_0^\infty \! h(x)f(x)\mathrm{d}x=\...
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43 views

Expected value word problem from Khan Academy

Hello all and thanks for taking the time to read this. I'm closing out the last few sections of statistics in Khan Academy, but there is a problem that is really bugging me. The problem reads like ...
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50 views

Expected value of difference of two order statistics

$X_1,X_2,..,X_n$ is a random sample from the random variable whose pdf is, \begin{align*} f(x)=\lambda e^{-\lambda(x-\mu)},\mu<x<\infty \end{align*} How can we find $E(X_{(2)}-X_{(1)})$, if $n=2?...
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38 views

How do we obtain E(exp(C+By-Y'aY))?

How can I prove this ? I did get $B-2uA$ by identifying the $a+by+cy^2= -1/2 (y-u)^2/\sigma^2$ But, I don't know how to get $(2a+\Sigma^{-1})^{-1} \dots$ Am I missing something? A little help would ...
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21 views

Expected value of a function of a multinomially distributed random variable

I have a scalar function, $g(x)$, where $x$ is an $n$-vector following a multinomial distribution with mass $f(x;p, N)$, for some probability-vector $p$, such that $\sum p_i=1$ and where $\sum x_i = N$...
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65 views

What is the minimum sample size for kaplan meier

I used the "survival" package in R to calculate a Kaplan Meier estimate for survival. An example of my output is like this: ...
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73 views

Approximate Order Statistics for lognormal variables

Are there any known formulas that approximate the expected value of the maximum of $N$ i.i.d. lognormal random variables? I am looking for something similar to: Approximate order statistics for ...
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23 views

expectation of conditional expectation

Given $(X,Y)$, 2-dimensional probability vector, and let $g: R^2 \rightarrow R, E[g(X,Y)^2 ] < \infty$ and $h:R \rightarrow R, E[h(X)^2] < \infty $, prove the following: $$E[h(X)\{g(X,Y)-E[g(X,...
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35 views

Why is the expected value of y written as E(y|x)?

The expected value of the simple linear regression model $y = \beta_0 + \beta_1x + \epsilon$ is typically written as $E(y|x) = \beta_0 + \beta_1x$. Why is it written as $E(y|x)$ instead of just $E(y)$?...
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31 views

Expectation of “mixed-variables” obtained from a N(0,1) variable

I have a variable $z$ which is normally distributed with zero mean and unit variance. I should derive $$ E[z^+]$$ $$ E[z^-]$$ $$E[z^+z^+]$$ $$E[z^-z^-]$$ $$E[z^+z^-]$$ where $z^+ =max(z, 0)$ and $z^- =...
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Stationarity of the TGARCH

I'm going through "GARCH models" by Francq and Zakoian (2010). They define the TGARCH(1,1) as $$\sigma_t = \omega + \beta_1 \sigma_{t-1} + \alpha_{1,+}\epsilon_{t-1}^+ - \alpha_{1,-}\epsilon_{t-1}^- $$...
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40 views

How to estimate an expected value of f(x,y) when x and y are random

So I have 3 sets of data. I'll call them x, y, and z (it's not a secret or anything what these variables are, I'm just trying not to distract from the question). x has bounds of 0 to 150 and is random ...
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1answer
65 views

Order Statistics, Expected Value of range, $E(X_{(n)}-X_{(1)})$

$X_1, X_2,...,X_n$ is a random sample from $U(0,\theta)$. Find $E(X_{(n)}-X_{(1)})$. I attempted this question by first finding the CDF of $X_{(n)}-X_{(1)}$ using the formula: $$F_{U}(u)= n\int_0^\...
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69 views

If X~Exp(λ), what is the expected value of Y=X²?

I am trying to compute this using the integral definition of expected value but I don't think I am doing it right as I am getting a very hard integral that I can not solve. When computing $\mathbb{...
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8 views

Coordinate Ascent for Variational Inference: Deriving Updates

I am working with the following model and am attempting to derivate coordinate ascent updates using mean field variational inference: Sample $p_X \sim Beta(\alpha_1, \alpha_2)$ Sample $p_Y \sim Beta(...
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50 views

Expectation of precision, recall, f1

Let $X^n$ be a sample of size $n$ drawn from a Bernoulli distribution with mean $\rho$. Let $Y^n$ be a sample of predictions drawn from another Bernoulli distribution with mean $\gamma$. It's easy to ...
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338 views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\...
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68 views

How do the means of $X^2$ and $X$ compare?

If $X$ has an exponential distribution with mean $\theta$, does $X^2$ have mean $\theta^2$? If not, how would I find the variance of $X^2$? I tried this: $$V(X^2) = E[X^4] - E[X^2]^2$$ But I'm ...
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70 views

How do I find the expected value of F(isher)-distribution

$E(F)=\int xf_{k,m}dx$ where $f_{k,m}(t) = \Gamma(t)=\frac{\Gamma((k+m)/2)}{\Gamma (k/2)\Gamma(m/2)}k^{k/2}m^{m/2}t^{k/2 - 1}(m+kt)^{-(k+m)/2}$. How do you find $E(F)$? Say you have to convert $x*f(k,...
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1answer
61 views

Expected time to get all four unique coupons [duplicate]

Envelopes are on sale for Rs. 30 each. Each envelope contains exactly one coupon, which can be one of four types with equal probability. Suppose you keep on buying envelopes and stop when you collect ...
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79 views

What is $V(X^t)$ for any $t$ when only $E(X)$ and $Var(X)$ are known and $X$ is assumed normal?

Summary I'm trying to calculate $Var(X^t)$ where $t$ is the number of periods using only the following known parameters: $E(X)$ and $Var(X)$. $X$ is a random variable and is the return factor $(1 + ...
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22 views

Distribution and Expected value

Table 1 1 -5 12 -4 14 -3 13 -2 9 -1 11 0 10 1 10 2 8 3 6 4 6 5 First column above is frequency, second is data. Sum is -22, Count is 100, Mean is ...
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59 views

Expected value of absolute difference of random variables

Given two continuous random variables X and Y with joint pdf f(x,y)=1 if 0<=X<=1 and 0<=y<1 I want to find E(|X-Y|) What I have done so far is to calculate marginal Fx and Fy ...
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99 views

Expectation of rational formula

I have two independent normal random variables $x$ & $y$ that are zero mean and unit variance. $a$ & $b$ are positive. I need to find the mean of $$z=\frac{ax^2y^2}{1 + bx^2}.$$ Any help ...
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How to find expectation of absolute values just from the data given below?

Suppose that $X$ and $Y$ are random variables such that $$E(X + Y ) = E(X - Y ) = 0 ;$$ $$\operatorname{Var}(X + Y ) = 3 ;$$ $$\operatorname{Var}(X - Y ) = 1.$$ (a) Evaluate $\operatorname{Cov}(X,...
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36 views

Confidence Intervals of a one sided truncated normal distribution

Assume $y\sim N(\mu,\sigma^2)$ it can be shown that UMP alpha ($\alpha$) level tests are derived from the $\alpha/2$ quantiles i.e. $z(\alpha/2)$ How can we find confidence intervals of a truncated ...
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35 views

Expectation of $\mathbb{E}(Tr(X^T A X))$ and $Var(Tr(X^T A X))$?

What is the expectation: $\mathbb{E}(Tr(X^T A X))$ and $Var(Tr(X^T A X))$ when $X_{i,j} \sim N(\mu, \sigma^2)$ and $X \in \mathbb{R}^{n \times k}$ where $n>k$ and $A$ is a given p.s.d matrix (not ...
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66 views

Expected number of groups of 3 consecutive wins in 200 rounds

If I know the probability of winning an individual round (while playing, say, a slot machine), call it $p_0$, what is the expected number of groups of 3 consecutive wins in 200 rounds? Or, to make it ...
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1answer
46 views

ELO rating for non-pairing sport + serious math

I was considering sport disciplines for which there are multiple players at the event but rather than playing against each other, they do stuff, are assigned points and their final position is based ...
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Justifying an early equation from *Introduction to Statistical Learning* [duplicate]

I'm self-studying Introduction to Statistical Learning. Page 19 of the book states the following: Consider a given estimate $\hat{f}$ and a set of predictors $X$, which yields the prediction $\hat{...
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26 views

Finding conditional expected value

Given that X and Y are two independent exponentially distributed random variables with parameters a and b respectively. let Z = max(X,Y) find E[X|Z] attempt: I found that: P(Z=X) = b/(a+b) and P(...
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69 views

Where do the default values in the Elo ratings formulas come from?

After doing some reading about the Elo ratings system, I am trying to implement one. I have some questions on the default values in the formulas. If player A has rating $r_a$ and player b has rating $...
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25 views

Expectation of the maximum of two correlated normal variables

I am curious what the derivation for the expectation of the max of two jointly normal random variables $X$ and $Y$ with correlation coefficient $\rho$. I could start with the following but the ...
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1answer
24 views

Measuring forecast accuracy of the conditional mean

Consider a dependent variable $y$, independent variables $x_1,\dotsc,x_K$, a model $$ y = X \beta + \varepsilon $$ and an estimated coefficient $\hat\beta$. If the model is assumed to be well ...
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14 views

Regressions and expected value

Assume I have $Y=\beta_0 + \beta_1*X_1+u_0$ and $Y=\alpha_0+\alpha_1*X_1+\alpha_2*X_1^2+u_1$ where $E[u_0|X]=E[u_1|X]=0$ When is it true that $\alpha_1=\beta_1$? I did a sort of reversal proof: ...
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43 views

Probability of at least one triangle in Erdos-Renyi graph

This is a well-known problem in random graph theory, where we show that if $X$ is the number of triangles in $G(V,E,p)$ with $p=o(\frac{1}{n})$, we can show that $$ P(X \geq 1) \geq 1-o(\frac{1}{n}) \...
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45 views

Derivation of Equation of Reducible and Irreducible Error [duplicate]

I am currently reading An Introduction to Statistical Learning by James, Witten, Hastie, and Tibshirani, and I am stuck on one of the leaps they take when defining reducible and irreducible error. ...