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

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28 views

Expectations of squared sum question

I can't seem to figure out why these expectations turn out the way they do, I am currently studying about the Fisher Information. If $X_1,X_2,...,X_n $ are all iid Poission($\lambda$) , then going ...
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1answer
21 views

Problem with Broyden update: Divide by a matrix?

I am implementing a maximum likelihood method (the EM algorithm) for which I'm using Broyden's method at each iteration. Here is the formula: $\Delta A = \frac{(\Delta \theta - A ...
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1answer
10 views

Variance with minimal MSE in normal distribution

Given $X_1,...,X_n$ ~ i.i.d. $N(\mu, \sigma^2)$ where the mean is unknown, let the estimator for $\sigma^2$ be $\hat{e} = p\sum_{i=1}^n(X_i-\overline{X})^2$ How do I choose $p$ so that this estimator ...
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2answers
39 views

Inverse of a mean, exponential distribution, expected value

Could you help me find the expected value of this random variable? Let $X_1, X_2, ... $ be independent identically exponentially distributed with parameter $\lambda$ random variables. What is the ...
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0answers
16 views

First order moment of multivariate Gaussian random vector

Let $X = (X_1,\dotsc, X_n)$ be a random vector distributed as a multivariate Gaussian with mean $0$ and covariance $\Sigma$. What is $\mathbb{E}[X_1\dots X_n]$?
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19 views

Width of Gaussian distribution from N trials of coin tossing

What is the width of the Gaussian distribution that is generated from performing $N$ trials of coin tossing? Example: In a trial of 1000 tosses of a coin, $P(H)=0.5$ and $P'(H)=0.5$, where $H$ refers ...
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0answers
24 views

Property of covariance of Normal random variable with an arbitrary function of that random variable

In the paper Sharpee, T., Rust, N.C., Bialek, W.: Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). I found the following claim ...
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32 views

Expected value of key presses

I came across this extremely strange problem that revolves around a piano. I'm not sure how to go about solving it because of the peculiarity. There are 9 notes on a mini piano, numbered from 1 to 9. ...
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0answers
19 views

Why moments of expectation are known as “moments”

I am studying moments of expectation, and seen the formulas for computing the moments. There is one thing I am not clear of, and not getting answer for that. Why moments are named as moments? To my ...
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2answers
33 views

Finding an unbiased estimator for a parameter, dicrete variable

Let $X : \Omega \to \mathbb{N}$ be a random variable. Define $p_i = P(X=i), \ \ i \in \mathbb{N}$. Find an unbiased and consistent estimator for $p_1$. I need to find an estimator $\alpha_n(X_1 + ...
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1answer
21 views

Setting up an expected value problem involving a Poisson distribution

I have the following problem: Let $N$ be the number of losses and let $$ T[N] = \begin{cases} \$0 & \text{ if }N = 0\\ \$10,000 & \text{ if }N = 1 \\ \$17,500 & ...
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0answers
18 views

Variance of sample variance

We are given $\{X_i \} $ iid random variables with $\mathbb{E}X_i = \mu$ and $D^2X_i < \infty$. I'm trying to compute $D^2(\sigma^2_n)$ where $$\sigma^2_n= \frac{1}{n} \sum_{i=1}^n (X_i - \mu)^2$$ ...
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0answers
18 views

Expectation of inverse of sum of random variables, exponential distribution

I have a question similar to this one: Expectation of inverse of sum of random variables only my variables have exponential distribution. So $X_1, X_2, ...$ are $iid$ with exponential distribution. ...
2
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1answer
48 views

Expected Payoffs

In simple setting consider revenue is dependent on variable $w$ which is uniformly distributed $[0,1]$. The revenue function is $wd$, where $d$ is development program. How to I get the expected ...
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2answers
73 views

For X,Y random variables, with pdfs that are symmetric around 0, does $V(X)\geq V(Y) \Rightarrow E(|X|)\geq E(|Y|)$?

I need to show the following thing. Consider two continuous random variables $X,Y$ which take values in $[-1,1]$ and are have pdf's that are symmetric around zero. How can I show that $V(X)\geq V(Y) ...
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1answer
19 views

Expected loss in regards to a question containing a continuous random variable with uniform distribution

I have a general question about a homework assignment that deals with a uniform distribution of a continuous variable. Here is the question (and the parts of the question): Suppose parking rules are ...
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2answers
42 views

Calculating the expected value of a random variable that's a function of a random variable

I am working on the following problem: I'm having a hard time putting all of this information together: The cost of the maintenance is $Z = X + Y$, where $X$ is the cost of the first machine and ...
2
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1answer
30 views

Expected Value and variance of a max randomized stocks

Hey guys I have been working on a probability and expected value/variance problem and the problem is: Each day the price of a stock in the market is a random number between 0 and 1 independently of ...
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3answers
16 views

Expectation of minimum discrete and continious random variable

I have two random variables, one is $X$ which has $\mathbb{P}(X=2)=\mathbb{P}(X=4)=\mathbb{P}(X=6)=1/3$, and one, call it $Y$, which is $unif(3,5)$ (continious) distributed. I want to compute ...
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1answer
41 views

Finding Variance

I am a little confused on how to go about finding different parts of the Variance of a random variable. Here is the question. A total of $n$ balls, numbered $1,.. n$, are put into $n$ urns, also ...
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41 views

Variance and Expected value of internet connection

I am working on a probability/statistics problem! The problem is as follows: Your internet connection is very poor. It constantly alternates between being functional for x minutes and being down for ...
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1answer
48 views

How to find $E[X^2\mid X+Y]$?

Suppose $X$ and $Y$ are independent Poisson random variables with rates $\lambda_1, \lambda_2$ respectively, then how would we go about calculating: $ E[X^2\mid X+Y] \text{ ?} $$
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1answer
37 views

Representing the probability as a recurrence equation

Introduction Suppose that you initially have an $n$-sided die with equal probability and you throw it then you will get a certain number $1< k \leq n$ then you throw a $k$-sided die. Continue ...
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20 views

Let $G$ be a random variable with $G=G(X,Y)$ where $G(x,y)=3x+y^2$. find $E(G)$

Let $G$ be a random variable with $G=G(X,Y)$ where $G(x,y)=3x+y^2$. Find $E(G)$ i know that $E(X)=E(Y)=1$ and $Var(X)=Var(Y)=1/2$ is this enough to answer the question? Im sure this is'nt too hard ...
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13 views

Explicit computation of a simple expectation

Let $N_t=P_t-\lambda t$ be the compensated Poission process. Has anyone seen either of the following expected values $$E\Big[\Big(\int_0^tf_s\,dN_s\Big)^k\Big]\quad \text{or}\quad ...
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2answers
31 views

Computing expectations by integrating survival function

I am trying to show that $\mathbb{E}(\text{max}(X-u,0))=\int_u^\infty(x-u)dF_X(x)=\int_{u}^{\infty}(1-F_X(x))dx$ for $X$ positive r.v. and $u>0$. I already know how to prove that ...
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1answer
52 views

Expectation of a linear combinations of iid standard normal, restricted to a halfspace

Let $u = (u_1, \ldots, u_n)\in\mathbb{R}^n$ be a unit vector in $\mathbb{R}^n$, $Y_i$ be i.i.d standard normal Is there any easy way to calculate $$\mathbb{E} \left[ 1_{\displaystyle \left\{ ...
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1answer
18 views

Finding a node in a full binary tree: expected number of comparisons

Consider a full binary search tree of height $k$ (the root is on level $1$ and the leaves on level $k$). By full I mean that all leaves are on level $k$ and level $k$ has exactly $2^{k-1}$ leaves. In ...
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25 views

Prove that the variance of a discrete random variable increases with a parameter

I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$. Let ...
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0answers
14 views

perfect coin is tossed n times. Let Sn denotes the number of heads obtained. What is the expectation of Sn?

The Problem is: A perfect coin is tossed n times. Let Sn denotes the number of heads obtained. What is the expectation of Sn? I got to E(Sn) = $\sum_{n=1}^{+\infty} \space\space\space Sn ...
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1answer
18 views

Finding the variance of X with a variable p between 0 and 1

Here is the full question (and note this is a question on my homework, but I just don't understand a portion of it): Suppose a random variable $X$ has the support of $S_X = \left\{0,1\right\}$ with ...
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1answer
63 views

Expected Value of the Maximum Number of Heads in n Flips

How would one go about finding the expected value of the maximum number of consecutive heads when flipping a coin $n$ times? For small $n$, it seems easy to brute-force it (i.e. when $n = 3$, the ...
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2answers
47 views

Variance of the Maximum Likelihood Estimator of the parameter of a Rayleigh distribution

I want to calculate the variance of the maximum likelihood estimator of a Rayleigh distribution using $N$ observations. The density probability function of this distribution is : $$ f(\sigma,y_i) = ...
4
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1answer
51 views

$E(X^2)=E(X)=1$. Find $E(X^{100}).$

$X$ is a random variable such that $E(X^2)=E(X)=1$. Find $E(X^{100}).$ My attempt: Assuming $X$ is discrete, we have $\sum x_i\mathbb P(X=x_i) = \sum x_i^2\mathbb P(X=x_i) = 1.$ We have something ...
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1answer
55 views

Variance of the number of r.v summed to fill certain capacity

Let us assume that we have a certain capacity T. We have an infinite number of random variables $X_1,X_2,\dots,$ where each $X_i$ is independent and has a particular pdf $P_i(X)$. And we have that ...
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2answers
29 views

Random variables and expectations

I am not even sure on how to get started on this: Let $k \ge 1$. (i) find a rv X for which $E|X^k|< \infty$ but $E|X^{k+1}|= \infty$ (ii) Is it possible to find a rv Y for which $E|Y^k|= \infty$ ...
3
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1answer
39 views

For i.i.d. random variables $X_1,\ldots,X_n$ find $\mathbb{E}\frac{\sum\limits_{i=1}^k X_i}{\sum\limits_{i=1}^n X_i}$

Let $X_1,\ldots,X_n$ be independent and identically distributed random variables and assume that the expectation of $X_1$ is finite. For $k \le n$ find: $$\mathbb{E}\frac{\sum\limits_{i=1}^k ...
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1answer
24 views

box with 2n cards, choose m.

This is a problem solved by bernoulli, and is given as an example in my textbook.But I'm having a little trouble understanding it. In a box containing 2n cards, there are 2 cards marked 1, 2 cards ...
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1answer
42 views

How to find conditional expectation $\mathbb E(X|X<M)$

Consider a random variable $X$ following the so-called folded normal distribution. That is, $X$ has density function $$ f_X(x) = \sqrt{\frac{2}{\pi\tau}}e^{-\frac{x^2}{2\tau}}, x>0. $$ ...
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1answer
50 views

Expectation and Convergence of Sum of Random Variables [closed]

Let $X_1, X_2, ...$ be a sequence of independent random variables with $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Let's now consider the sum $S_n=\sum_{k=1}^{n} X_k$. I need to show three ...
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2answers
96 views

Entropy upper bound inequality for Sub-Gaussian Random Variable

We say that the random variable $Z$ is $\sigma^2$-subGaussian if $\mathbb{E} \exp(tZ) \leq \exp(t^2\sigma^2)/2$. Define the $(x\log x)$-entropy (or simply the entropy) of a nonnegative random ...
2
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1answer
25 views

A variation of the coupon collectors problem

The problem goes as following: Let there be $n$ coupons, and $X_i$ be the random variable whose value is $1$ if coupon $i$ is collected during the first $n$ draws, and $0$ otherwise. What is the ...
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1answer
40 views

Expected value question

My teacher gave the following question as a practice question for the exam... I was just wondering if someone could check if my answer is correct: A group of n ≥ 3 people is sitting at a round table, ...
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0answers
18 views

probability problem using Chebyshev's inequality

Suppose that a die has its "3" side changed to a "2". The problem is to first find a lower bound on the probability $P[3\leq X \leq 4]$ using Chebyshev's inequality. Then if we roll the die $n$ ...
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2answers
43 views

Is the expected value of a random variable always constant?

$ \DeclareMathOperator\Cov{Cov} \DeclareMathOperator\E{E} $Can I say that the expected value of a random variable is always a constant? For some context, I am trying to wrap my head around the ...
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0answers
54 views

expected value minimum of bivariate normal distribution

Let $X,Y$ be jointly normal with density $f(x,y)=\dfrac{1}{2\pi\sqrt{1-\rho^2}}\exp(-\dfrac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2))$. Let $Z=\min(X,Y)$. Show that $E[Z]=\sqrt{\dfrac{1-\rho}{\pi}}$ and ...
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1answer
53 views

Coin Flipping Game - Expected Number of Tails

Can someone help me with this problem? In this game, let $S_{t}$ denote your earnings at time $t$. Your initial earnings is one dollar ($S_{0} = 1$). For each subsequent time, $t = 1, 2, ..$, flip a ...
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0answers
22 views

Entropy of AR(1) and AR(2) model

Does anyone know any suitable papers or knowledge themselves on the steps involved in calculating how the entropy of a AR(1) or AR(2) time series model? For example, for an AR(1) process of the form: ...
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0answers
26 views

Is this the right way to do it?

There are 40 questions. For each question, there are 5 options of which only one option is correct. 3 points are awarded for each correct answer, and 1 mark is deducted for each wrong answer. For ...
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0answers
14 views

MSE for the Method of moments estimator of variance

would appreciate some help here please - Question: Find the MSE for the MOM estimator of the variance $\hat{\sigma^{2}} = \frac{n-1}{n}S^{2}$ based on a random sample from a normal distribution. My ...