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

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Moments of quadratic forms

$x=(x_1,...,x_T)'$ is a $T\times1$ random vector, where $x_t, t=1,..., T$, is a stationary process with mean zero and finite fourth moments. $A$ is a $T\times T$ symmetric constant matrix. How to find ...
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What is wrong with this logic based on a geometric distribution?

Problem #10 from A Collection of Dice Problems by Matthew M. Conroy, is: Suppose we can roll a $6$-sided die up to $n$ times. At any point we can stop and that roll becomes our score. The goal is ...
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28 views

Expected value of trials to obtain a red ball in a box of white balls.

I have a problem that involves a box containg N balls, one of which is red and the rest of which (N-1) are white. The question involves finding the expected value and variance for the number of trials ...
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Is $P(n) = \frac{a n }{b}$ or $\frac{(a+1) n}{b + 1}$?

I investigated Some random data and I was a bit confused. Could be Mathematical coincidence but i'm not sure. Consider the integers $1,2,3,...,a$ Randomly Pick $b$ dinstinct element out of them. ...
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128 views

Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.

I am currently reading a "pop-science" book on statistical fallacies. On page 36 the authors discuss how events can cluster around certain locations by chance. The authors exemplify this by a $6*6$ ...
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Notation of expectation and random variables

I'm trying to understand the notation used at p18 of The Elements of Statistical Learning. I suspect errors in notation. What do the authors mean and, if any notational errors, what would be the ...
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21 views

Expected value of a discrete random variable

Ok guys, I have a problem with proving this result... I have a random variable $Z$ that can take the values $[1, 2, 3]$ with probability $[\pi_1, \pi_2, \pi_3]$. How can I prove that $\mathbb{E}[Z]=2$ ...
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What is the average number of draws (2 cards per draw with shuffles in between) before I had seen all 52 cards in the deck?

On average, how many times would I have to draw two cards from a deck (replacing and shuffling between each draw) before I had seen each of 52 cards in the deck? The process is: Draw the top two ...
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21 views

Expectation of exponential of integral of absolute value of Brownian motion

Sorry about all the "of"s in the title... here's my problem: I want to compute the expected value of $$ \exp\bigg\{ C \int_0^t |W_s|ds\bigg\} $$ where $W$ is a Brownian motion and $C$ is a positive ...
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inequality involving expectation of the maximum

For $X_i \sim$ i.i.d with cdf $F_x$, and $\forall c \in \mathbb R$, then, letting $M_n$ denote the maximum observation $$ M_n \le c+ \sum_i^n (X_i - c) \mathbb I(X_i > c) $$ I proved this by ...
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Taylor expansion of an expectation

Ok guys, I'm reading a book and I'm not getting quite well a concept. If I have to expand $U'(Y_0(1+r_i))$ around $Y_0(1+r_f)$, why I get this: ...
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Poisson Process Question - arrival times

Car crashes at a traffic stop arrive follows a Poisson Process with rate lambda = 2/hr. (a) Expected amount of time until the 2nd car crash arrives (b) P(See at most 2 car crashes during rush hour) ...
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43 views

Question about Poisson process and arrival times

Problem: On any given day you receive mail in mailbox with probability $p$. Assume whether mail is put in the mailbox or not is independent each day. If the neighbor receive mail in his mailbox ...
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52 views

Coupon collector problem doubts

The Coupon Collector problem off Wikipedia: Suppose that there is an urn of $n$ different coupons, from which coupons are being collected, equally likely, with replacement. How many coupons do you ...
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How to compute the expectation of a projection matrix?

$X$ is a $T\times k$ random matrix with finite second moments, how to compute the expectation of the projection matrix $E[I_T-X(X'X)^{-1}X']$? (Assume that $X'X$ is positive definite.)
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expected value of final x using the following algorithm

It has a bit of pseudo code so i'll try to explain x = 0 for i from 1 to n: if random() > 1/4: x = x + 5 else: x = x - 1 the probability ...
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What is meant by Expectation or Expected value of a Random Variable?

Probability: In terms of Relative frequency. $S$: Sample Space of an experiment $E$: Experiment performed. For each event $E$ of sample space $S$, we define $n(E)$ : no. of times in first $n$ ...
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expectation of product of sums of normally distr. r.v.

Let $Z_1$ and $Z_2$ be i.i.d. standard normally distributed. $X_1=Z_1+Z_2$ and $X_2=Z_1-Z_2$. Apparantly E[|$X_1|*|X_2|$] = E$[|Z_1|*|Z_2|]$. Why?
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Expectation of Independent Variables Equals Zero?

Given $n$ independent random variables, $X_1, X_2, ..., X_n$ , each having a normal distribution, why is it that the following expectation holds? $$E[(X_i - \mu)(X_j - \mu)] = 0$$ where $i \neq j$ ...
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Computing expectation of: $\small E\left[f(Z,U)e^{\frac{V^2-(V+W)^2}{2}} \right] $

Suppose we have three mutually independent random variables $U,V,W$ where $W \sim \mathcal{N}(0,1)$, $V \sim \mathcal{N}(0,c)$ and $E[U]=0$, $E[U^2]=1$. Lets define $Z=U+V+W$. Can we compute (or ...
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10 red cars, 10 blue cars, 10 green cars are distributed randomly in a line. What is the expected number of times a red car precedes a blue car?

We can apply linearity of expectation to this to make it easier. This means we reduce the problem to finding the expectation that the red car i precedes the blue car j. This is just the probability of ...
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42 views

Expected number of jacks drawn given that you draw cards till you draw all 4 kings?

I don't understand how to solve this. Basically define J as our random variable such that J: {0,1,2,3,4}. To solve this, we need to know the probability of getting e.g., 0 jacks given that we draw ...
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38 views

Computing expected value of exponential

Let $U$ and $V$ be two independent random variables where $E[U]=$ and $E[U^2]=1$ and where $V$ is standard Gaussian. We also let $W=U+V$. How to compute the following expectation or find an an upper ...
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Prove that $E(|(X+Y)(X-Y)|) \leq 2\sqrt{1-\rho²}$, where $\rho$ is correlation.

For two random variables $X,Y$ with mean $0$ and variance $1$, their correlation is $\rho$. We have to prove that $$E(|(X+Y)(X-Y)|) \leq 2\sqrt{1-\rho^2}.$$ But, I can't understand how the $\rho$ ...
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Determining the MVUE of $ f(x;\theta) = \theta^x (1-\theta)$.

The Statement of the Problem: Let $X_1, X_2, ... , X_n$ be a random sample from $$ f(x;\theta) = \theta^x (1-\theta) \quad x = 0,1,2,... $$ (a) Find the ML estimator of $\theta$. (b) Show that $T ...
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29 views

Is there something wrong with this Probability Density Function?

Problem: A certain software company uses a certain software to check for errors on any of the programs it builds and then discards the software if the errors found exceed a certain number. Given that ...
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Breaking a stick randomly at two points: Expected value of the largest piece. [duplicate]

If a stick of unit length is broken randomly at two points, to make 3 pieces of stick. What is the expected value of the largest stick. Is there an elegant solution to this problem? Thanks.
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Expected value from popping bubbles

This is a fairly simple math problem from a programming challenge but I'm having trouble wrapping my head around it. What follows is a paraphrase of the problem: ...
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Hash Table expectation question

A hash table is a commonly used data structure in computer science, allowing for fast information retrieval. For example, suppose we want to store some people’s phone numbers. Assume that no two of ...
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I want to find PDF by differentiating CDF and then from PDF, expected values of the following problem. .

Three light bulbs have independent exponentially distributed lifetimes with a common parameter $\lambda$. What is the probability distributed function and expected value of the time until the last ...
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Cauchy-Schwarz Inequality for Expectation of N random variables

I have seen many books that state and prove the Cauchy-Schwarz inequality for two, positive-valued random variables $X$ and $Y$ with bounded expectation as \begin{equation} E[XY]^2\le E[X^2]E[Y^2]. ...
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Determining the efficiency of $2 \overline X$.

Let $X$ be a random variable with pdf $$ f\left(x;\theta\right) = \left\{ \begin{array}{lr} \frac{3\theta^3}{(x+\theta)^4} & \text{if } 0<x<\infty \text{ and } 0 < \theta ...
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How to show that $P(X_{n} \geq n$ $i.o.) = 0$, given $E(X_{i}) = 0$ and $E((X_{i})^{2})=1$ for $i=1,2,3…$

I'm struggling with the following problem (Exercise 4.5.16 in Rosenthal's probability book): Let $X_{1}, X_{2},...$ be defined jointly on some probability space, with $E(X_{i}) = 0$ and ...
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How do I compute such theoretical expectations?

Let a spider take a random walk on an $n$-vertexes complete graph, starting from vertex number $1$. (I guess it has no significance.). Given the spider visits vertex $2$ before he visits vertex $3$, ...
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Infinite expected value?

An acquaintance of mine proposed a scenario. Imagine parents who ground their child. Initially, the grounding is for 5 days, but for every day the child misbehaves while they're grounded, the ...
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Implementation of EM algorithm for Gaussian Mixture Models using Matlab

Using the EM algorithm, I want to train a Gaussian Mixture model using four components on a given dataset. The set is three dimensional and contains 300 samples. The problem is that after about 6 ...
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Expected value definitions

Let $X : \Omega \to \mathbb{R}$ be a discrete random variable in a discrete probability space with countable sample space $\Omega$. Let $P(\omega)$ be the probability of an outcome $\omega \in ...
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Expectations of stopping times in general

I have a very basic question: So for a stopping time $\tau$ with $E(\tau)<\infty$ we have $E(\tau)=\sum_{n=0}^\infty P(\tau>n)$, right? Why is that? Thanks!
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How long before the prey can escape?

I've (sort of) come across the following problem in my research. The actual scenario is a little abstract to explain, so I'm rephrasing the problem in terms of a predator/prey scenario. I'm tagging ...
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43 views

Finding the expected value and variance of $X$

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the expected value and variance of $X$ and I know that: cumulative distribution function: ...
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49 views

compute temporal average of $\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)$

assuming that $\Phi$ is uniformly distributed over $(0,2\pi)$ compute: $$E[\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)]$$ I have solved the problem as continues: $$\begin{align} ...
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Expected value using indicator variables

Randomly, $k$ distinguishable balls are placed into $n$ distinguishable boxes, with all possibilities equally likely. Find the expected number of empty boxes. PROPOSED SOLUTION: Let $I_j$ be the ...
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what is the difference between statiscal averagre and average?

I'm reading a book on synthetic aperture radar and it is said that: The term $\sigma^{\circ}$ is the averaged radar cross section per unit area, also called the scattering coefficient or ...
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Expected Shortfall alternative definition

Define: $$q_\alpha(F_L)=F^{\leftarrow}(\alpha)=\inf\lbrace{x\in \mathbb{R}\mid F_L(x)\geq \alpha\rbrace}=VaR_\alpha(L)$$ I want to prove that: $$ES_\alpha = ...
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29 views

Expected Value of Two Random Variables

X is a random variable with a probability density function $f(x)$, g(x,y) is a function of two variables one of them is the random variable. I have \begin{equation} \int_{-\infty}^{\infty} ...
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Relationship between minimizing a conditional variance and a covariance

We are working with discrete-time stochastic processes. Let $v_k$ be a $\mathcal F_k$-predictable process, and let $X_k, \eta_k$ be $\mathcal F_k$-adapted processes. Define $V_k = v_kX_k+\eta_k$ and ...
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44 views

Probability Uniform Distribution Set Up Integral

Consider a $1$ meter stick and suppose you break it into two pieces $X$ meters from the end, where $X \sim \operatorname{Unif}(0,1)$. What is the expected length of the longer piece (after it is ...
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39 views

Simple Question about Monotone Convergence Theorem

Suppose we have a sequence of (discrete) random variables $X_0, X_1, \dotsc$ over $E$ and $A \subseteq E$. Let $Y$ be some other random variable. Moreover, let $Z$ be a random variable with values in ...
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Uniform distribution and expectation

Let $U \sim \mathrm{Unif}(0,1)$, $X=U^2$ and $Y=e^X$. Compute $E[Y]$ (leave answer as an integral). So essentially we need to compute $E[e^{U^2}]$? I am a little confused how to approach this problem? ...
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67 views

Conditional Expectation: Sum inside or outside of expectation?

Let $X,Y$ be some discrete random variables with $Y$ taking values in $\mathbb{N}$ and consider $\mathbb{E}[X]$. Since it is sometimes easier to consider the expectation conditioned on a certain ...