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

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1answer
23 views

How is derived it? E(exp(-bdt))=E(1-bdt) [on hold]

I have following transformation: E(exp(-bdt))=E(1-bdt). But I don'tknow how is it derived? Could anyone clarify? Thank you!
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1answer
31 views

Application Problem: Expectation and Variance of Compound Poisson Process

I am solving the following: Let $Y1, Y2,…$ be a random sample from $\Gamma(p,a)$ distribution, where p and a are positive real numbers. $Y$ is damage in thousands of dollars caused to a car in an ...
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1answer
14 views

When do I use Law of total variance?

For example, at the beginning of doing this problem (http://math.illinoisstate.edu/krzysio/3-6-10-KO-Exercise.pdf), I was thinking of using $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, ...
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1answer
12 views

Expectation of geometric summation of exponentail random variables

Let $\{\tilde x_i, i = 1,2,\ldots\}$ be a sequence of iid exponentially distributed random variables with parameter $\lambda$ and let $\tilde n$ be a geometrically distributed random variable with ...
0
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1answer
28 views

independence of random variable

Suppose we have $2$ Independent random variables $X$ AND $Y$. Let $f(X)$ and $g(Y)$ are functions of those $2$ random variables. 1.) my question can we say that the functions $g(X)$ AND $f(Y)$ are ...
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0answers
27 views

How would you understand the notation $\operatorname E(\zeta \mid x)$.

Let ($\Omega=X\times Y, 2^{\Omega}, \operatorname{P})$ be a discrete probability space, so $\Omega$ consists of pairs $(x,y)$. Let $\zeta$ be a random variable $\Omega\rightarrow\mathbb{R}$ on that ...
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1answer
14 views

How to prove that expectation is the integral of survival function? [duplicate]

I am trying to prove that $E[X] = \int_0^{\infty} P (X > x)$ I have started like below: $$\text{E}[X] = \int_{0}^{\infty}x f_{X}(x) dx $$ $$ = \int_{0}^{\infty}\int_0^x dy f_{X}(x) dx $$ ...
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2answers
59 views

An exam`s points dilemma

On July 2 I have an exam, in this exam will be 40 questions in test with 5 variants of answer for each question. For each correct answer will be given +1 point. For each incorrect answer will be ...
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1answer
43 views

Basic question concerning conditional expectation (of a non-mathematician)

Let $(X_i)_{i \geq 1}$ and $\tau \geq 1$ be independent random variables with $\mathbb{E}[X_i]=\mu$ for all $i \geq 1$. Moreover, let $S_k:= \sum_{i=1}^k X_i$. I want to show that ...
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0answers
31 views

Why is $\sum_{k=1}^{\infty}\mathbb{E}[\mathbb{1}(T=k)]=\sum_{k=0}^{\infty} k \mathbb{P}[T=k]$

Let $T$ be a non-negative random variable. Why is it true that $$\sum_{k=1}^{\infty}\mathbb{E}[\mathbb{1}(T=k)]=\sum_{k=0}^{\infty} k \mathbb{P}[T=k]$$ According to me it would make sense that ...
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1answer
39 views

Trouble finding the expected value of a random variable

Suppose that we have a procedure A that we run once and it returns as a result either success or ...
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1answer
38 views

Find the expected value of a game [closed]

It costs $\$10$ to play a game. You have a $15\%$ chance of winning. You collect $\$30$ if you win. Otherwise you lose your $\$10$. Find the expected value.
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1answer
40 views

How can I find the density of $E[X\mid Y]$ when $(X,Y)$ is gaussian

I was tying to prove the following: Given $(X,Y)$ a centered gaussian vector in $\mathbb{R}^2$ with the following covariance matrix $$ \Sigma = \begin{bmatrix} \sigma^2_x & \sigma_{x,y} \\ ...
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2answers
102 views

What distribution has $X^n$ if $X$ is normal distributed?

Let $X$ be a random variable with mean $0$ and variance $\sigma ^2$, i.e. $X \sim \mathcal{N}(0, \sigma ^2)$.What is the distribution of $Y= X^n$, $n \in \mathbb {N}.$ ? I know what distributribution ...
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0answers
16 views

Expectation calculation at infinty

While reading a paper, i came across the following Expectations: Given that the $E\left\{e^2_{n-i-1}e^2_{n-j-1}\right\}=E\left\{e^2_{n-i-1}\right\}E\left\{e^2_{n-j-1}\right\}$ for $i\neq j$.\ Then ...
2
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1answer
25 views

Computing Distribution of Conditional Expectation of Gaussian RV

I am trying to compute distribution of the following random variable \begin{align*} E[(X-E[X|Y])^2|Y] \end{align*} where $X \sim \mathcal{N}(0,\sigma^2_x)$ and $Z \sim \mathcal{N}(0,\sigma^2_Z)$ where ...
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1answer
32 views

what does E[$c^x$] mean in probability

Hello I'm self studying probability this summer and I would like your help to clarify me on this question. Let x be such that $P(X=1)=p=1-P(X=-1)$. Find $c≠1$ such that $E[c^x]=1$ Can anyone tell ...
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1answer
34 views

Calculating $E[(X+Y)^2]$ where $X$ and $Y$ are dependent Bernouli RVs.

Given: P(X) = 1/4, P(Y) = 1/3, P(XY) = 1/8 (so not independent since P(XY) does not equal P(X)*P(Y) Since these are Bernoulli RVs, we also have E(X) = 1/4, E(X^2) = 1/4, E(Y) = 1/3, E(Y^2) = 1/3, ...
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1answer
16 views

Terminology: Expected Value, Expectation, Expectation Value

According to [Wikipedia::Expected Value] expected value and expectation are correct terms for the first moment of a random variable. What about expectation value? I have heard and read this term ...
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1answer
26 views

Calculating $ \mathbb E \left[e^{-\mu W_T } 1_\left( {\min W_t \leq a} \right) \right]$ for a Wiener process

Let $W_t$ be a standard Wiener process, $a$ some real number, and $\chi (x)$ the indicator function. I am trying to calculate the following expectation: $$ \mathbb E \left[e^{-\mu W_T } \chi \left( ...
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2answers
38 views

expectation of lgamma of gamma distribution

Is there a closed-form expression for $E[\log(\Gamma(X))]$, where $X \sim Gamma(k, \theta)$? Edit: Note the gamma function inside the log. Edit 2: If there's no closed-form expression, is there a ...
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1answer
36 views

Upper bounds on $E[{\rm var}^2(X|Y)]$

I am looking for an upper bound on the quantity \begin{align*} E[{ \rm var}^2(X|Y)] \end{align*} where ${\rm var}(X|Y)=E[(X-E[X|Y])^2|Y]$. Getting a lower bound is rather easy using Jensen's ...
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1answer
24 views

If $\lim_{n \rightarrow \infty} E(X_t| \mathcal{F}_{t-n}) = 0 $ then $E(X_t) = 0$?

Suppose I have a sequence of random variables $X_t$ adapted to a filtration $\mathcal{F}$ when is it true that if $\lim_{n \rightarrow \infty} E(X_t| \mathcal{F}_{t-n}) = 0 $ then $E(X_t) = 0$ ? ...
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2answers
26 views

Expectation value of number of drawings of increasing sequences of labelled balls from an urn.

An urn contains $n$ balls, labelled from $1$ to $n$. A sequence of drawings with re-insertion is made, until the drawn ball is labelled with a number which is less than or equal to the number of ...
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0answers
38 views

Secretary Problem - Optimal algorithm for expected value of candidate.

I recently encountered the secretary problem and there are essentiall two problems: Maximizing the probability of choosing the best candidate. Gnedin proved in his paper ...
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1answer
19 views

Expected value of the maximum of poisson distributed variable and 0 [closed]

I want to find the expected value of max{S-X,0}, where X is Poisson(lambda) distributed and S is a fixed number. I am looking for the notation with integrals.
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1answer
33 views

Application Problem: Random Sums of Random Variables and Correlation

I am trying to answer the following: The number of traffic accidents per year at a given intersection follows a Poisson(10, 000)-distribution. The number of deaths per accident follows a ...
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1answer
26 views

finding expected value using conditional expectation

i have stumbled upon the following problem and have absolutely no idea how to approach it.the problem is as follows- Suppose we have a circle, centered at the origin, with a circumference of $12$ ...
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1answer
27 views

If the $(n-1)$th moment exists does the $n$th moment necessarily exist?

Let's suppose the distribution is unknown but that the second moment is known to be finite. Doesn't this imply that the distribution should fall off exponentially fast and therefore higher moments ...
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0answers
27 views

Gibbs sampling truncation for contrastive divergence

I am following Yoshua Bengio's Learning Deep Architectures for AI and at page 31 there is a phrase that confuses me. Starting by lemma 7.1 in the same page: Lemma 7.1. Consider the Gibbs chain ...
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0answers
14 views

What is the distribution of the distance if the points are uniformally distributed?

I have two points say $\mathbf{p_1} = (x_1, y_1)$ and $\mathbf{p_2}= (x_2, y_2)$ which are uniformally distributed with parameter 0, 1, i.e., $\mathcal{U}(0, 1)$. These two points are generated inside ...
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1answer
37 views

How to calculate $E(\sin^2X)$

If $X \sim N(0,1)$ then calculate $E(\sin^2X)$ I understand that $0 < \sin^2x<1$. So the expectation exists. I proceed as $E(\sin^2X)= \int_{-\infty}^{\infty}\sin^2xf(x)\,dx=2 ...
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1answer
25 views

Expectation of quotient of linear combinations of independent standard normal random variables

Let $a, b, c, d, e, f$ be complex numbers with nonnegative real parts and nonnegative imaginary parts, and let $X_{1}, X_{2}, X_{3}, X_{4}$ be independent standard normal random variables. How can I ...
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1answer
19 views

Are RV having same exp. value and covariance already have the same distribution?

Let $(X_1, ..., X_n), (Y_1, ... , Y_n)$ be random variables. $X_i$ has the same distribution as $Y_i$ for all $i$. $\forall i, j: Cov(X_i, X_j) = Cov(Y_i, Y_j)$ Do $(X_1, ..., X_n)$ and $(Y_1, .., ...
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1answer
43 views

A miscomprehension of how to compute $E[X^2]$.

During a question concerning the Birthday Paradox, given P is the number of pairs of people with the same birthday in a group of 20 people and assuming there are 365 possible birth-days, I am to ...
2
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1answer
27 views

Convergence in distribution with finite mean

I'm preparing myself for the final exam of my graduate Probability Theory course and was stuck with another one of the exercises our professor gave us. Let $X_n, n=1,2,\ldots,$ and $X$ be nonnegative ...
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2answers
25 views

Construct a 98% confidence interval for the unknown expectation µ.

Today, I started to learn about Confidence intervals for the mean. Unfortunately, I don't understand the following exercise of the book: You are given a dataset that may be considered a realization of ...
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1answer
15 views

Expectation of Integral of Brownian Motion

I'm working through some stochastic analysis problems at the moment and I've come across a problem that is a bit tricky (to me) - does anyone know how to calculate this expecation? I'm not sure what ...
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1answer
26 views

Numerically solving equations with expectations

I have a equation $\mathbb{E}_\theta f(x,\theta)=a$, where $\theta$ is a vector real random variable with a known distribution, $a$ is a real constant, $x$ is a real (can be vector valued) variable. ...
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1answer
34 views

Expectation of $b^T \operatorname{sign}(Ab)$.

I'm trying to compute the expectation of: $$b^T \operatorname{sign}(Ab)$$ Where $b$ is a $n\times1$ vector of independent Bernoulli random variables: $$P(b_i = 1) = 0.5,\quad P(b_i = -1) = 0.5$$ and ...
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2answers
77 views

prove : if E(X) doesn't exist $E(x^2)$ too doesn't exist.

$E(X^2) $ exists implies $\int x^2 f_X(x) \ dx < \infty$ now from the property of Riemann Integral $\int|x| f_X(x) \ dx \le \int x^2 f_X(x) \ dx $ . hence, existence of $E(X^2)$ implies ...
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1answer
25 views

Application Problem of Expected Value of Posterior Distribution

I am trying to understand the following: Suppose that the number of people who visit the grocery store on any given day is Poisson($\lambda$) and the parameter of the Poisson distributed has a ...
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1answer
34 views

Looking for expectation of the number of substrings

The question is formulated as follows: if we are given $n$ random binary strings of length $n$, what is the expectation of the number of substrings they have in common? Sounds pretty simple, but if ...
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1answer
33 views

Find the mean and variance of $V_n=\frac{1}{n}\sum_{i=1}^n(X_i-u)^2$

Suppose that $X_1,X_2,...,X_n$ is a random sample from a distribution with mean $\mu$ and variance $\sigma^2$. Suppose also that $v:=\mathbb{E}[(X_1-\mu)^4]<\infty$. Find the mean and variance of ...
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0answers
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Expectation of an absolute exponential generalized beta 2(EGB2) distributed variable

For a bachelor thesis i'm trying to impose a new GARCH model using an non-linear exponentioal GARCH model with an underlying Generalized beta distribution of the second kind(introduced by ...
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1answer
43 views

Posterior Distribution and Expected Value of a Coin Toss where Probability of Heads is a Random Variable

I am trying to solve the following: Suppose X is the number of times a coin is tossed until a heads is observed. Let Y denoted the probability of observing heads and assume $f_Y(y)=ky^2$, ie the ...
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0answers
34 views

expectation value of independent random variables

In the statistics lecture that I'm attending, the professor once used the following: $X, Y$ random variables and i.i.d., then $$\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$$ I was trying to see an ...
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1answer
75 views

Brainteaser: Player A has £1, Player B £99. They flip a coin: variance

I am trying to figure out a question concerning a problem found here: Brainteaser: Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. Expected number of games ...
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0answers
22 views

What is the expected number of cards I should buy to have all of them? [duplicate]

Background:My little brother collects cards and tries to complete a book of them, In order to depress him a little I'm trying to compute the number of he need to buy in order to collect all of ...
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2answers
36 views

Random variables with common mean, variance and pairwise correlation

Hi I'm currently working through past exam questions and am stuck with the following question: Random variables $X_1$, $X_2$ and $X_3$ are identically distributed, with common mean $\mu$, common ...