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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$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
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Prove $P(T > \infty) = 0$

From Williams' Probability with Martingales How exactly do we know $P(T > \infty) = 0$? If the partial sums are bounded, then no matter how many terms in the sequence we add, they won't ...
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9 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
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Prove $|M_n^T| \le c + K$ [on hold]

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
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4 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
319 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
66 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
55 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
56 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
75 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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1answer
48 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 ...
2
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1answer
62 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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19 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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1answer
25 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
63 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 ...
4
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1answer
34 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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24 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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60 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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20 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
20 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
35 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
30 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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1answer
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
19 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
138 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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1answer
35 views

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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1answer
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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Expectation of Error 2k squares?

There is a regression model $$Y_n= X_n \beta + \epsilon_n $$ where, $X_n$ is a $n \times p_n $ matrix, E($\epsilon_n$) = 0, and Var($\epsilon_n$) = $\sigma^2$ How can I figure out E( ...
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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
43 views

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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29 views

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
87 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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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 ...
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Approximating the conditional expectation in simulations

I am simulating stock returns, which are governed by the following equations $r_t = \mu + \delta r_{t-1} + \varepsilon_t$ $\sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma^2_{t-1}$ ...
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25 views

Taylor Series to Find Expected and Variance [duplicate]

Let g(X) be a specified function of X. The first order Taylor approximation of g(X) in a neighborhood of y is g(X) squilly = g(y) + g'(y)(X-y) The right hand side is a linear function of X. Using ...
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1answer
74 views

predicted vs expected values using lmer in R

I am running a multilevel model with several interactions and a binary treatment. To summarize the model I would like to compute the first differences between treatment (1) and control group (0) using ...
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1answer
35 views

Expected value of Uniform distribution

Suppose $X$ is an uniform random variable: $X \sim U(a,b)$. I know how to compute $E(X)$, but what if I want to compute: $E(X^\gamma)$ where $\gamma > 0$?
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Calculating the Title odds

Given a set of expected values within an event. How do you calculate the probability of each of them occuring the most? i.e. Who will win the league? total goals scored in a season , given expected ...
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What is the expectation for a multivariate normal distribution?

Assuming we have a multivariate normal distribution Y∼N(μ,Σ) with known μ,Σ, then, how to compute the expectation for this multivariate gaussian, is that exactly equals to μ? Thanks.
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Expectations of kernel density estimate

Suppose $X_{1},..., X_{n}$ are independent and identically distributed according to the probability density function $f$. Let ...