For questions about the expectation of a random variable: computations, upper/lower bounds, etc.

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15 views

Distributional convergence and expectations

I'm struggling with the following problem. Let $X_n$ be a sequence of non-negative random variables which are finite almost surely and all with expectation 1. Assume they converge in distribution ...
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22 views

Detailed explanation needed for basic query regarding expectation

I need to find the expectation of following random variable $$g=[\log_2(\frac{1+x}{1+y})]^+$$ where $[x]^+=max(x,0)$ and both $x,y$ variables depend on variable $z$. I know the conditional pdf's and ...
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0answers
40 views

Expectation and Variance of an Estimator

Imagene following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+y^2}{...
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3answers
30 views

How to find E(x) and Var(x) in this specific continuous probability distribution.

I've got into some confusion on continuous probability distributions, and everything related to it. This is the problem: Problem Image. I assume from the sketch that pdf is $f(x) = x$ for values of $x$...
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0answers
30 views

Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
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0answers
40 views
+50

Expected score from threshold with number deletions

We play a game where a sequence of $n$ numbers is drawn uniformly from $[0,1]$, and we need to set a threshold $0\leq a\leq 1$. For every number that is at least our threshold, we get $a$ points but ...
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2answers
21 views

Does this hold in every case, and if only this one, why? Expectation, mean of random variable.

Characteristic function of random variable $X$ let us denote as $f_X(t)$ and $EX$ it's mean or expectation. Does the following hold in all cases, because it keeps coming up and I don't know why it is ...
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1answer
35 views

Is $\lim_{x\to\infty} x\overline{F}(x)=0$?

With partial integration I wanted to prove that for non-negative random variable with CDF F(x) holds $$ \int_0^{\infty}\overline{F}(x)dx=E[X]. $$ Here is $\overline{F}(x)= 1-F(x)$. I got this far $$ \...
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4answers
92 views

Find the value of $\mathbb{E}(X_1+X_2+\ldots+X_N)$ of i.i.d random variables $X_i$s.

Let $ X_1,X_2,X_3 ,…$ be a sequence of i.i.d. random variables with mean $1$. If $N$ is a geometric random variable with the probability mass function $\mathbb{P}(N=k)=\dfrac{1}{2^k}$; $k=1,2,3,\...
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0answers
40 views

Bounding $E[X]E[e^{-X}]$ for a non-negative random variable with finite moments [closed]

Let $X$ be a random variable such that $X \geq 0$ and $\exists \varepsilon > 0: E[e^{X \varepsilon}] < \infty$. Is it true that $\exists$ a real number $B < \infty$ such that $E[X]E[e^{-X}] \...
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1answer
41 views

Why is $f_x(Ax + b) = f_x(x)$?

Let $x \in \mathbb{R}^n$ be random vector, $A \in \mathbb{R}^{m \times n}$ be a matrix and $b \in \mathbb{R}^m$ be a vector. Now I should proof that the expected value is linear: $$\mathbb{E}(Ax + b)...
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3answers
81 views

How can I compute $\mathbb{E}[Z^4]$ where $Z\sim N(0,1)$

Let $Z\sim N(0,1)$ and $Y=a+bZ+cZ^2$. I want to compute the variance of $Y$. This is what I did: $$\operatorname{Var}(Y)=0+b^2\operatorname{Var}(Z)+c^2\operatorname{Var}(Z^2)=b^2+c^2\operatorname{Var}...
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2answers
33 views

How to compute variance of a conditional expectation and vice versa

I am trying to use the law of total variance which is $$\operatorname{Var}(X) = \text{Var}(E(X\mid Y)) + E(\operatorname{Var}(X\mid Y))$$ But I honestly have no idea how to compute either one of ...
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0answers
38 views

What is E(X^a)? [closed]

In terms of expected value, is there a formula for $E(X^a)$, such that a is any real number? If not, how does one do so knowing the distribution of X, using the formula $E(X) = \sum xp_x(x)$?
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4answers
82 views

$E(X)$ versus $E(X|Y)$

Why is $E(X)$ considered a constant but $E(X|Y)$ considered a random variable? Seems like confusing notation since I'd assume the latter is a fixed constant "the expected value of random variable $X$ ...
2
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1answer
120 views
+100

Expected score from threshold

We play a game where a sequence of $n$ numbers is drawn uniformly from $[0,1]$, and we need to set a threshold $0\leq a\leq 1$. For every number that is at least our threshold, we get $a$ points, ...
1
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3answers
67 views

Law of total expectation?

Apparently $E[X] = E[E[X\mid Y]]$ but I don't understand what this really means. I looked at https://en.wikipedia.org/wiki/Law_of_total_expectation but need another explanation. Isn't this the same ...
4
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1answer
57 views

Proving $\text{Var}(X) = E[X^2] - (E[X])^2$

I want to understand something about the derivation of $\text{Var}(X) = E[X^2] - (E[X])^2$ Variance is defined as the expected squared difference between a random variable and the mean (expected ...
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0answers
30 views

expectation over poisson point process needed

I need the result of following problem $$E_z[\exp(-a(\frac{1}{z})^b)]$$ where $a>0,b>0$ and $z$ is defined as below (I do not know if there is other simpler way of defining $z$ but if somebody ...
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1answer
22 views

Least amount of repetitions s.t. probability greater than 1/2

Assume that for a formula $F$ over $n$ variables, there are exactly $k$ allocations that satisfy it. How many random samples from the set $\{0,1\}^n$ are necessary to find an allocation satisfying the ...
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0answers
29 views

Exchanging supremum and conditional expectation

I've come across a problem which seems similar to this but quite different and can't find a way of going around it. I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and ...
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1answer
43 views

Conditional expectation of a product of random variables

I have two independent continuous random variables $X$ and $Y$ with pdf's : $f(x)$ and $f(y)$ cdf's : $F(x)$ and $F(y)$ a constant $a$ I am trying to express using the given pdf/cdf functions the ...
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1answer
40 views

Expectation of absolute random variables with mean 1 and standard deviation 1

For a random variable $\gamma \sim \mathcal{N}(\mu,\sigma)$ , were is $ \mathcal{N}$ is the normal distribution. What is the way to calculate the following: $ \mathbb{E}[|\gamma|] = ? $ And ...
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1answer
26 views

Expectation of infimum of asymmetric 1D random walk greater than -$\infty$

I'm reading Durrett's book on Probability and in the example of the asymmetric 1D random walk with $P(X_1=1)=p>1/2$, when trying to compute the expectation of the hitting time $T_{b}:=\inf\{n: S_n=...
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2answers
30 views

Expectation of the sum of two cards without replacement = Expectation of the sum of two cards with replacement

We have $10$ cards numbered from $1$ to $10$. We pick two cards among them. What is the expected value of the sum of these two cards ? I have solved this question the hard way using the law of total ...
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0answers
13 views

Expectation of $Y=X\mathbf 1_{X>t}$ in terms of the CCDF $P(Y>x)$

I have random variable $Y=XU(X-t)$ (here $U(X)$ is the unit step function, $Y$ has non-negative support and depends on other random variable $X$) which has a CCDF $P(Y>t)$. I want to write the ...
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2answers
29 views

Fine E(4X^2+4X+1)

So I have the following tables $$ \left[ \begin{array}{c|ccc} x&-3&6&9\\ f(x)&\frac{1}{6}&\frac{1}{2}&\frac{1}{3} \end{array} \right] $$ I am tasked to ...
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2answers
253 views

Expected value of the zeros of random polynomials of degree two

Let $a_1,a_0$ be i.i.d. real random variables with uniform distribution in $[-1,1]$. I'm interested in the random zeros of the polynomial $$p(x) = x^2 + a_1x + a_0. $$ One thing (between many) thing ...
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2answers
43 views

$\mathbb{E}[|X|^n] < +\infty \implies \mathbb{E}[|X|^k] < +\infty, k \leq n$

Show that if $\mathbb{E}[|X|^n] < +\infty$, then $\mathbb{E}[|X|^k] < +\infty, \forall k \leq n$. I guess I have to apply Hölder Inequality, but I was not able to find out what $p$ and $q$ are ...
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24 views

situation when Monte Carlo method cannot be used

This is an example from notes I don't understand why we should think about the $E(x)$ and $var(x)$ first?
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1answer
43 views

Finding the limit $\lim_{t\to ∞} \mathbb{E}[R_t]$ of an SDE

I have the SDE $$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$ In this equation, $R_0 = r$ in which $r > 0$ Can someone please help me find the $\lim_{t\to ∞} \mathbb{E}[R_t]$? Thus far I have ...
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1answer
42 views

Computing the expectation value of a stochastic process

I have a stochastic differential equation for which I have solved the process X$_t$. The SDE is as follows: $$ dX_t = \left( r\mu X_t + \frac{r(r-1)} 2 \sigma^2 X_t \right) \, dt + r\sigma X_t\,dB_t, ...
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40 views

On the distribution and the moments of $\max\{1/\sqrt{U_1},…,1/\sqrt{U_n}\}$, where $(U_k)$ is i.i.d. uniform on $(0,1)$

Let $U_1,U_2,...$ denote an i.i.d. sequence of random variables with the uniform distribution on $[0,1]$. For every integer $n\geq1$, we set $M_n = \max\{1/\sqrt{U_1},...,1/\sqrt{U_n}\}$. a) Compute ...
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0answers
31 views

Issues regarding my take on proving $E(X) = \lambda$, where $X\sim Poisson(\lambda)$

My proof: Let $X\sim \mathrm{Poisson}(\lambda)$. Then $$f_{\Tiny{X}}(x) = \frac{\lambda^x}{x!} e^{-\lambda}.$$ Thus, $E(X) = \sum_{x=0}^{\infty} x f_{\Tiny{X}}(x) = \sum_{x=0}^{\infty} x \frac{\lambda^...
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1answer
35 views

How to deduce the expectation of a stochastic equation [closed]

I am having a difficult time deducing the expectation, $\mathbb{E}[R_t]$, of the following stochastic equation: $$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$ $R_0 = r$, with $r > 0$. Please help me ...
2
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3answers
27 views

Hypergeometric Random Variable Expectation

In a binomial experiment we know that every trial is is independent and that the probability of success, $p$ is the same in every trial. This also means that the expected value of any individual trial ...
2
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1answer
48 views

Expectation of $|X-Y|$ when a coin is thrown six times

If a fair coin is thrown six times. Let $X =$ number of heads and $Y = 6-X =$ number of tails. What is $E|X-Y|?$ I was able to come up with this table, but I am not sure if this is correct or not and ...
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1answer
38 views

Distribution of Expectation function into a $|X-Y|$

We know that $E(X+Y) = E(X) + E(Y)$. But why is $E|X-Y|$ $\ne$ $E|X| - E|Y|?$
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1answer
10 views

Expectation of Bivariate Distributions

I know very little about probability and I was searching for the expected value of bivariate distributions, but all I could find was the expected value of a real-valued function of the distribution. ...
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14 views

Average Area of Convex Hull of N points in Unit Hypercube

Suppose we randomly pick $N$ points inside the unit hypercube in $\mathbb{R}^n$, and form their convex hull. What is the expected value of the volume of the convex hull? For example, in the case $N=n=...
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1answer
56 views

If $X \sim Exp(1)$ and $Y \sim Exp(1)$, prove $(\frac{X}{Y}, Y)$ is continuos without using the change of variables theorem. [closed]

I've been thinking about this problem for a while and I'm not sure which way to go. Let $X$ and $Y$ be two independent random variables with exponential distribution of parameter 1. Let $U = \frac{X}...
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1answer
19 views

Expectation of a random variable and an indicator function [closed]

Suppose you have a random variable $X$, and an event $A$. How do you evaluate the expectation $\mathbb E[X\ \mathbb{I}_{A}]$?
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1answer
29 views

Minimum of maximum of independent variables

I'm trying to find the probability distribution and expected value of the minimum of maximums of a combination of random variables. For example, say $$X_1 \sim \mathrm{Exp}(\text{rate}=\lambda_1), ...
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1answer
32 views

Closed Form E[exp(x'Ax)]

Is there a (general) closed form available for the following expression? $$\mathbb{E}\left[e^{x^{T}Ax}\right]$$ Where: $$x=\left\{ x_{1},x_{2},...,x_{N}\right\} \sim\mathcal{N}\left(0,\varSigma_{N}\...
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0answers
31 views

If the SD of the speed of vehicle is $8.8$ kmph and the mean speed is $3.3$ kmph, find the coefficient of speed.

If the standard deviation of the speed of vehicle on a highway is $8.8$ kmph and the mean speed of the vehicles is $3.3$ kmph, find the coefficient of speed. $SD =8.8$ Variance=$(SD)^2$ E($x$)=$3.3$...
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0answers
47 views

How can this double summation be solved?

I have to calculate the following expectation $$\mathbb{E}\left[\left(\frac1M\sum\limits_{i=1}^MX(i-n_1-M)\right)\left(\frac1M\sum\limits_{j=1}^MX(j-n_2-M)\right)\right]$$ where $M$, $n_1$ and $n_2$ ...
3
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1answer
78 views

Mean number of fixed points of random function from $\{1,2,3,\ldots,n\}$ to $\{1,2,3,\ldots,n\}$

Expected value (mean) tends to confuse me lately. Let $ \Omega$ be the discrete space of all functions $ \omega:\{1,2,3,\dots,n\} \rightarrow\{1,2,3,\dots,n\}$, $\mbox{Pr}$ is uniformly ...
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0answers
45 views

Best estimator for 'expected goals' in a soccer game with known outcome

Consider a soccer game between team $A$ and an irrelevant opponent. We know the outcome of the game: Team $A$ scored $G_A$ goals, had $ST_A$ shots on target and $S_A$ shots. Assume that the expected ...
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0answers
28 views

Conditional Expectation with linearity

Solving $E(X)$ $$=E(X-a+a)$$ (By linearity) $$=E(X-a)+E(a)$$ $$=E(X-a)+E(a)$$ $$=E(X-a)+a$$ Does this hold for all probability distributions? The place where this seems counter-intuitive to me is ...
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2answers
99 views

How to understand this integral result?

I was reading this page on Wikipedia: Birthday Attack I can understand up until how to approximate the minimal number of attempts for a given probability $$n(p; H) \approx \sqrt{2H \log \frac 1{1-p}...