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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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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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 = ...
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45 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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+50

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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1answer
22 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: ...
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34 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 ...
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18 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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1answer
39 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
58 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)= ...
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1answer
68 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 ...
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7 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 ...
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47 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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1answer
335 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^*} ...
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2answers
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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1answer
56 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 ...
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1answer
58 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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2answers
77 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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1answer
21 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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54 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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3answers
97 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 ...
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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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1answer
30 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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2answers
64 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
41 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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16 views

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 ...
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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 ...
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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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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
21 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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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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1answer
37 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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1answer
40 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. ...
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29 views

Expectation of a function of a binomial distribution

I have a question that is: Given n iid Bernoulli(p) distributions: $X_1, X_2, \ldots, X_n$ and $S_n=\sum X_i$. Find $E[(S_n-np)^3]$. Hint: $S_n-np= \sum (X_i-p)$. So far, I have gotten that $S_n$ is ...
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Expectations of white noise

Under what conditions is E[XY]= E[X] E[Y] ? If X and Y re white noise then why would this equation hold true?
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How many data points to check to validate confidence in an algorithm

Say I have an algorithm which scrapes n = 3,000 sets of data from the web. I want to know whether the scraping is successful. Therefore I will closely check the results of a number of these data. I ...
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1answer
20 views

Expectation of discrete random variable

Give a sequence of random variables $x_1,..,x_n$ with $x_n$ having a density of: $$f_N(x) = \begin{cases} \frac{2N-1}{3N};x=1\\ 1/3;x=1+\frac{1}{N+1} \\ \frac{1}{3N};x=2\end{cases}$$ What would be ...
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2answers
143 views

Is frequentist statistics concerned with expectation? [closed]

Frequentist statistics sees probability as the expectation of the value. Expectation is the long-term average. Do Frequentists interpret probability as the expected value for that parameter? EDIT: ...
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What is the expected distance from the mean of a Gaussian? [duplicate]

For $p(x) = N(x|\mu,\sigma)$, what is $E[|x-\mu|]$? Is it the standard deviation $\sigma$?
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Characteristic function issue

As mentioned in a previous post, I've been trying to work through ALL of the problems in Jacod and Protter's Probability Essentials. The following problem has been giving me issues: Let $Z \sim ...
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Working out the expectation of a function of iid random variables

I have found the maximum likelihood estimator $\hat{\sigma}$ of a iid r.vs $X_1, ..., X_n$ which all have normal distribution with known mean $\mu$ and unknown variance $\sigma^2$. So $\hat{\sigma}$ ...
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35 views

Binomial and Poisson issues (Jacod and Potter)

I've been reading through Probability Essentials by Jacod and Potter (2nd edition). I'm on a voyage to do every single exercise in the book. The following problems I am unsure of is as such: 5.11) ...
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21 views

Quantile Regression Expected Value

I know this is probably painfully simple, but can someone help me with the following? $\textbf{Model:}$ $y=x'\beta(u)$ where $u|x\text{~}Uniform\,[0,1]$ and for any $x,\, x'\beta(\tau)$ is a ...
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2answers
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Mean Squared Error as Reducible and Irreducible Component

I am having problem with a basic proof . I want to decompose Mean Square Error into Reducible and Irreducible parts as shown below, but I cannot go from the step 2 to step 3. \begin{align} \mathbb ...
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Expectation of t distribution

Find $E(T^{2r})$ of Student's t-distribution. My teacher has told me the answer.... it's something like $$ \frac{n^r \Gamma(r+(1/2)) \Gamma(n/2-r)}{\Gamma(1/2) \Gamma(n/2)} $$ From this I calculated ...
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1answer
88 views

How does one find the expected value of $\text{E}(XY)$ when $\text{Cov}(X,Y)$ is not zero? [closed]

What do I need to determine $\text{E}(XY)$ when $\text{Cov}(X,Y) \neq 0$?
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31 views

Widgets and boxes problem: expectation and variance. Why is this wrong?

I'm taking the MITx: 6.041x Introduction to Probability - The Science of Uncertainty class to sharpen my probability skills. In one of the problems, the solution I came up with diverged from the ...
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The expected value of a bunch of dependent events

Suppose we have pairs of number randomly selected in $(0,1)$. $$ (a_1, b_1) \rightarrow (a_2, b_2) \rightarrow ... $$ We start at $t = 1$ and continue to $t = 2$, and so on. At each step $t$ we ...