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

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How to compute P(|X - E_Y[h(y)]| < c)?

Consider the discrete random variable $Y$, the continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, ...
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28 views

vase with blue and red balls

At first I hope this is not a duplicate post. I tried to find it but I have not found it. I hope that someone could help me with understanding the exercise. This question is about a vase with r red ...
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1answer
6 views

Autocovariance between two dependent variables?

I got a simple question that I really don't have understood 100% yet We let $\{X_t\}_{t=-\infty}^{+\infty}$ be stationary AR(1) process given by: $X_t + 0.25X_{t-1} = Z_t$, where $\{Z_t\}$ is ...
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40 views

Find the density of a ratio of random variables

$X$ has density $2x, 0 < x < 1,$ and $Y$ has density $1/10$ over $0 < y < 10$. $X$ and $Y$ are independent. I have to find (a) density of $Y/X$ (b) $E[Y/X]$ (c) $E[Y^2/X]$ I let $Z=Y/X,$ ...
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21 views

Two related question, in one. Same topic: Dispersion..

$1.$ Prove: If $X_1,X_2,X_3,\ldots,X_n$ are independent random variables then: $$D\left(\sum_{i=1}^n X_i\right)=\sum_{i=1}^n D(X_i)$$ Proof: Because of independence we have: $$D(\sum_{i=1}^n ...
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14 views

Given the matrix representation what is the expectation value

For a particle with spin $\frac{3}{2}$, construct the matrix representation for $S_z, S_x$ and $S_y$. If the particle is in an eigenstate of $S_z$, what is $\langle S_x\rangle$ and $\langle ...
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34 views

We write down the date of each person's birthday we meet (say Feb 29. doesn't exist).

Random Variable $X$ is the number on persons we met til we wrote down every date in a year. Find the expected value of $X$. Find $E(X)$- expected value. From this example I can definitely understand ...
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10 views

Poisson Distribution Optimization Problem

A retailer buys $n$ cookies and has to pay $\zeta_1$ for each. He wants to sell them for a price of $\zeta_2$ (with $0$ < $\zeta_1$ < $\zeta_2$). Let X be a random variable which states, how ...
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25 views

Interpretation of composite of random variable

Let $~f:[0,1] \to[0,\infty]$ be a measurable function bounded by $c \in \mathbb R$. Let $X_1,X_2,..,X_n \sim i.i.d ~\text{uniform}(0,1)$. How do I interpret the following statement: $$ Var(f \circ ...
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30 views

In a box there are $M_1$ balls numbered 1, $M_2$ numbered 2… $M_N$.

In a box there are $M_1$ balls numbered 1, $M_2$ numbered 2... $M_N$. From the box $n$ balls are drawn without returns. Find the mathematical expectation of the number of numbers that are not drawn. ...
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1answer
39 views

Calculating complicated expectation

I need to calculate $\operatorname{E}( X_2 \mid X_1=x, Y=y)$, where $Y=\max\{X_2,X_3\}$ and joint density of $X_1$, $X_2$ and $X_3$ is given by: ...
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1answer
15 views

If the second moments are uniformly bounded, does $Y_n$ converge in $L^2$?

Let $\{X_n\}$ be a pairwise uncorrelated sequence of random variables such that there exists a fixed constant $c>0$ such that $E(X_n^2)\leq c$ for all $n\geq1$. Does it imply that for any ...
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1answer
98 views

Sum of i.i.d. random variables and finding an upper bound

Problem: Suppose that $(X_i)_{i\in\mathbb{N}^+}$ is a sequence of i.i.d. random variables. For some $n\in\mathbb{N}^+$, let $S_n=\sum_{i=1}^n X_i$. Furthermore, let $a$ be a positive constant, and ...
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2answers
52 views

9 room probability and expected value

I got the following question: In a house with 9 rooms. There is 1 mouse that is looking for some food. This can be found in 2 rooms, but there are also 2 cats, these are in different rooms. When the ...
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Expectation of an interval

Given $g(\theta) := Pr\{X\leq\theta\leq Y\}$ with $Y\geq X$, what is $E[Z]$ where $Z:= Y-X$ ? Also $X{\not\perp}Y$ Progress: $$X\leq\theta\leq Y\Rightarrow \{Z \geq \theta-X\}\cap \{Z\geq\ ...
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The copy-problem : Does any block of digits appear at least twice?

Suppose, $N$ random digits have been generated. Let $X$ be the largest natural number with the following property : There are natural numbers $i$ and $j$ with $i+X-1<j$ , such that the digits $i$ ...
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2answers
39 views

Computing expectation exercises; using linearity of expectation and iterator random variables

Disclaimer: This is homework that is overdue by, but I do want to understand it and get through it, so any hints or guidance is appreciated This is for an algorithms class currently dealing with ...
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3answers
50 views

Change the order of expectation

Sorry this might be a silly question, but I'm kind of confused and really want to make sure I'm correct. Let $v_1,v_2,\dots,v_n$ be $n$ i.i.d. random variables with the same range of ...
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1answer
19 views

Graph Theory: Conditional Expected Value of Product of two Random Variables

Consider a graph with $n$ vertices, where each edge between any two vertices is independently drawn with probability $p$. Let $D_i$ be the degree of vertex $i$. What is $E[D_i \cdot D_j]$? Here is ...
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Computing MMSE and conditional expectation

Suppose we have three independent, zero mean, finite variance random variables $V,W,Z$ and where $W,Z$ are Gaussian random variables. These random variables form a new random variable $Y$ ...
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31 views

bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \sim P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one efficiently ...
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1answer
48 views

Prove absolute sum expectation

I have encountered the following problem, could someone provide me some hints on how to solve it? Assume that the sequence $(X_n)$ is i.i.d. with mean $0$ and variance $1$. For every $n\ge1$, let ...
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21 views

Conditional expectation of continuous RV

Let $X$ be random variable and $f$ it's density. How can one calculate $E(X\vert X<a)$? From definition we have: $$E(X\vert X<a)=\frac{E\left(X \mathbb{1}_{\{X<a\}}\right)}{P(X<a)}$$ Is ...
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33 views

What is the Difference in the Average and the Mathematical Expectation in the following Problem

Suppose that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students, for a total of 1000 students. The average class size is simply ...
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1answer
20 views

Proof for Mean of Geometric Distribution

I am studying the proof for the mean of the Geometric Distribution http://www.math.uah.edu/stat/bernoulli/Geometric.html (The first arrow on Point No. 8 on the first page). It seems to be an ...
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10 views

Identify functional form of expectation value

I have the equation: $g(z) = {\int_{\mathbb{R}^n}}\;f(x)\exp(-c(x)^Tz)dx = \mathbb{E}_X[\exp(-c(X)^Tz)]$ where $c:\mathbb{R}^n \to \mathbb{R}_+^m$, $z \in \mathbb{R}_+^m$ and $f(x)$ is a ...
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2answers
35 views

Expected time to pick up n things when I can drop them each round?

Why Can't I Hold All These Limes? Suppose I wanted to hold $n$ limes. Each time step, I pick up a lime. But limes are hard to hold, so I also have a probability $p$ to drop a lime on each step which ...
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3answers
44 views

Finding Variance and Expectation of Boolean Variable

Below is the joint distribution of Boolean random variables X1, X2 and X3. How do I find variance and expectation of X2? I understand that variance is "average of squares of difference from mean ...
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44 views

What allows us to divide a random variable into multiple ones?

I can't wrap my head around the solution presented for this problem: Suppose a trial has a success probability $p$, let $X$ be the random variable for the number of trials it takes to stop at $r$ ...
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1answer
26 views

Computing $E[X 1_{ \{ |Y|<1 \}}]$ [closed]

Am looking for any suggestions on how to compute \begin{align*} E[X \ 1_{ \{ |Y|<1 \}}] \end{align*} where $X$ and $Y$ are finite variance random variable assume zero mean. In my setting $Y=X+W$ ...
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27 views

find the power of a random process?

I know all the steps expect the last step i don't know how to evaluate the integral can someone show me the step that lead to the answer to be A^2/2
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28 views

Expected value of a random variable drawn from joint distribution, separated into parts

Suppose we have three random variables $X$, $Y$, and $Z$ that are drawn from a joint distribution $F(X,Y,Z)$ with joint density $f(X,Y,Z)$. I would like to write out the expected value of $X$ ...
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9 views

What is the expectation of semi-fixed-points in a random permutation?

1<=i<=n is a semi-fixed point if: |π(i)-i| <= 1 with π of {1...n} What is the expectation of semi-fixed point?
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23 views

Derivative of an Expectation over an Indicator function

I have a small question on how to compute the derivative of an expectation when an indicator function gets in the way. Let $x$ be a random variable. We are interested in computing the derivative wrt A ...
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9 views

Finding the correct distribution

Consider the independent stochasts $X$ and $Y$ that take the values $\pm 1$ with an equal chance (i.e. $1/2$). I came to the following distribution: $f_{X}(x)=(1/2)^{x}*(1/2)^{1-x}=1/2, x=\pm 1$ and ...
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Why $\mathbb E[S_T\boldsymbol 1_{\{T\leq n\}}]+\mathbb E[S_n\boldsymbol 1_{\{T>n\}}]=\mathbb E[S_{T\wedge n}]$?

Let $(S_n)$ a martingale refer to $(X_n)$ and let $T$ a stopping time. Prove that $$\mathbb E[S_{T\wedge n}]=\mathbb E[S_1],$$ where $a\wedge b:=\min\{a,b\}$. I have proved that $$\mathbb ...
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2answers
55 views

Expected value of max of three numbers

This is a combo problem that a friend came up with some time ago, and recently showed to me. He claims he solved it when it first occurred to him, but can no longer remember the solution, and neither ...
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1answer
45 views

If $\mathbb{E}[X] = 0$, then $P[X \geq \lambda] \leq \frac{\sigma^2}{\sigma^2 + \lambda^2}$

Say $X$ is a real random variable, and its expected value is $\mathbb{E}[X] = 0$. Denote the variance $\operatorname{Var}[X] = \sigma^2$. Show that $P[X \geq \lambda] \leq \frac{\sigma^2}{\sigma^2 + ...
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2answers
38 views

Show that MLE of $\lambda = \frac{n-T_n}{S_n+cT_n}$

$X_i$ are i.i.d exponential, mean $\lambda^{-1}$ for $1 \leq i \leq n$ and, the values are measured such that $X_i = c$ if $X_i \geq c$ and $X_i$ otherwise. Show that MLE of $\lambda = ...
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36 views

How to calculate a population mean for a normal distribution

This is for homework, but I'm a bit confused on how I can find $E(X_i) = \mu$ given a normal distribution. The question is as follows: In a farm, let $X$ denote the number of fruits harvested in a ...
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1answer
14 views

Expectation of trigonometric functions involving random variables.

This is more a formulation question. I need help making a sales pitch (lol). I am working on an practical engineering problem where I encounter functions of the form: $\cos(\phi + d_\phi)$, $ ...
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1answer
11 views

Which distribution satisfy expectation of square root of summing of square of coefficients is equal to summing coefficients? [closed]

Draw $C_1$, $C_2$, ... $C_n$ from some distribution, for any $\lambda_i>0$, we hope to get \begin{equation} \underset {C_1,C_2,\cdots, C_n}{\mathbb{E} } \left[ \sqrt{\lambda_1^2 C_1+ \lambda_2^2 ...
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1answer
47 views

Central Limit Theorem prove an expectation could be wrong

I really need your help. Here's the problem: A rookie is brought to a baseball club on the assumption that he will have a .300 batting average. (Batting average is the ratio of the number of hits to ...
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20 views

$E(E(X_{n+1}| X_0…X_n)) = E(X_n)$

Using Adam's Law to prove that a martingale has constant mean, we have $$E(E(X_{n+1}| X_0...X_n)) = E(X_n)$$ Why is the left side equal to $ E(X_{n+1})$? and not also $ E(X_{n})$
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Show that $EX_1 1_A \geq 0$ given A is an event $\left(\sum\limits_1^n X_i>0 \right)$

Let A be the event $\left(\sum\limits_1^n X_i>0 \right)$ Show that $EX_1 \mathbb{1}_A \geq 0$ Previous part: the first two parts of the question which I solved were to show how two ...
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51 views

Find $E[X+2Y|Z]$

$X,Y$ are independent standard normal. Let $W=X+Y$, $Z=X-Y$. Find $E[X+2Y|Z]$ Attempt: $E[X+2Y|Z=z] = E[X+2Y|X-Y=z] = E[Y+z+2Y] = 3E[Y]+Z = Z$ Is this correct?
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Formula for $E(X^4)$ as integral of complementary CDF of random variable $X$

So the question I have is Let S be a non-negative random variable. By writing the probability as an expectation and using Fubini's theorm, show that $ES^4=\int_0^\infty4t^3P(S>t)dt$ so I found ...
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1answer
38 views

Expected nof children “at least one boy and at least one girl, with boy older than girl”

A couple decides to keep having children until Cond1: they have at least one boy and at least one girl, Cond2: with boy older than girl and then stop. Assume they never have twins, that the ...
2
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1answer
19 views

How to determine the expectation of the square of a binomial collection

I'm trying to find how to express the expectation of the square of a collection of binomial measurements. If we have a collection: $$A = a_1 + a_2 + \cdots + a_n$$ The expectation of $A$ is the sum ...
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2answers
31 views

Deriving the variance of a binomial distribution

I know that the variance of a binomial distribution is the number of trials multiplied by the variance of each trial, but I'm not seeing the derivation of this. Here's my logic so far: For each trial ...