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

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Proportional probability of payouts with defined expected value.

Assume we have a lottery with payouts like this $(2,3,5)$ So you buy a ticket and you can win a pot which will multiply your ticket price by the numbers written ahead.The organizer expects a margin ...
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1answer
20 views

Expectation of product of iid random variables limited by stopping time

Let $X_1, X_2, \cdots$ be i.i.d. such that $X_i > 0$ and $\mathbb E[X_i]=1$ and consider $\mathbb F = \{\mathcal F_n\}_{n\ge 1}$ to be the discrete filtration. Denote $Y_n = \prod\limits_{i=1}^n ...
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17 views

Expectation of an exponent of a random variable

Suppose that $X \geq 0$ is distributed according to some distribution $F$. What can be said about $E[e^{-r X}]$? I.e. is there a way to express this expectation only in terms of some characteristics ...
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12 views

Reduction of Two Independent Random Variables in Quadratic Form

Consider the $n \times 1$ random vector $\mathbf{x}$ and the $p \times 1$ random vector $\mathbf{y}$. The vectors are independent of each other, and $\mathbf{y}$ has an expected value of zero. I want ...
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2answers
13 views

Find the moments of a binomial conditioned on a binomial?

Suppose that $Y$ has the binomial distribution, $Bin(20, 0.25)$ and conditioned on $Y$, a random variable $X$ that has the binomial distribution, $Bin(Y, 0.5)$. How can I derive the $k$th moment of ...
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2answers
234 views

Infinite expectation implies infinite random variable?

Let $X$ be a nonnegative integer-valued random variable depending on a parameter $L$. Assume that $E[X] \rightarrow \infty$ as $L\rightarrow \infty$. Does this imply that for any $k$, $P(X > k)$ ...
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1answer
9 views

Expectation of the minimum of two $\mathcal U(0, 1)$ r.v.'s conditional on it being greater than or equal to some value

Let $X_1, X_2$ be i.i.d. $\mathcal U(0, 1)$ (continuous) r.v.'s, and let $0 \le R \le 1$ be some number. What is $\mathbb E[\min(X_1, X_2) \mid \min(X_1, X_2) \ge R]$? My attempt: Let $Y = ...
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33 views

Expected Value Given pdf [on hold]

Suppose that fifteen observations are chosen at random from the pdf $ f_Y(y)=3y^2$, 0≤ y ≤1. Let $X$ denote the number that lie in the interval $[$1/2$ , 1]$. Find E(X), where E(X) is the expected ...
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19 views

Expectation of Conditional expectation over incorrect distribution

We want to compute the following quantity \begin{align} \int E[V|W=u] f_U(u) du \end{align} For random variables $ V,W,U$ where $V$ and $U$ are independent and $U$ is absolutely continuous r.v. Is ...
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3answers
435 views

Expected value problem with cars on a highway

There is a very long, straight highway with $N$ cars placed somewhere along it, randomly. The highway is only one lane, so the cars can’t pass each other. Each car is going in the same direction, ...
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23 views

Is the intuition correct?

Let $X$ be a random variable such that $E|X|<\infty$ and $$P\left(X\ge\frac{1}{2}+x\right)=P\left(X\le\frac{1}{2}-x\right)\ \forall x\in \mathbb{R}.$$ Then $E(X)=\frac{1}{2}$ and ...
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0answers
46 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
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1answer
21 views

Stopping time and the Martingale stopping theorem.

According to the book that I am reading, A nonnegative, integer valued random variable T is a stopping time for the sequence {$Z_{n},n\geqslant0$} if the event T = n depends only on the value of ...
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15 views

Limit to Expectation: $ - \lim_{N \rightarrow \infty } \frac{1}{N} \sum_{n=1}^{N} \frac{\partial}{\partial \theta} \ln p(x_n|\theta)$

When I was reading "Pattern Recognition and Machine Learning", I come across following equation (Equation 2.134, Chapter 2):$$ - \lim_{N \rightarrow \infty } \frac{1}{N} \sum_{n=1}^{N} ...
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23 views

Conditional expectation of another expectation expression.

What is the intuition and the proof behind the given below expression where $U,V,W$ are random variables: $E[V | W]$ = $E[E[V | U,W] | W]$ I know that $E[V | W]$ can be treated as a random variable ...
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1answer
33 views

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ , what is the mean of $X$?

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ for $x \in [0,1]$$ and $f(x) = 0$ for $x \notin [0,1]$. Then, the mean of $X$ is $\frac 12$ $\frac 1{\sqrt2}$ $\frac 13$ ...
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24 views

If $X$, $Y$ are random variables such that $E(X\mid Y=y)$ is constant for all $y$, then show that $E(XY)=E(X)E(Y)$ [i.e.,$\text{Cov}(X,Y)=0$] [on hold]

If $X, Y$ are random variables such that $E(X\mid Y=y)$ is constant for all $y,$ then show that $$E(XY)=E(X)E(Y)\qquad \text{[i.e. Cov}(X,Y)=0]$$
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260 views

Finding the expected value in the given problem.

It is given that a monkey types on a 26-letter keyboard with all the keys as lowercase English alphabets. Each letter is chosen independently and uniformly at random. If the monkey types 1,000,000 ...
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2answers
41 views

Find $\text{Var}(N)$ where $P(N = n|Y = y)$ is $\text{Possion}(y)$; $Y$ is a gamma with parameters $(r,\lambda)$

The question is as follows: Suppose that the conditional distribution of $N$, given that $Y = y$, is Poisson with mean $y$. Further suppose that $Y$ is a gamma random variable with parameters ...
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1answer
26 views

Expectations and densities

Because of an article that I'm trying to understand, I've come up with the folowing question: Suppose we have $f:(\mathbb{R}_{\geq 0})^2 \rightarrow \mathbb{R}_{\geq 0} $ , $\ X,Y\geq 0 \ \ $ ...
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0answers
62 views

Proof that Derivative of Expected Value is Zero (Using Differentiation show Unconditional Expectation is Constant)

If the expected value of a distribution is constant, it means its derivative with respect to the values it can take must be zero. I was wondering if there is a rigorous proof of the same. Steps Tried ...
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1answer
21 views

Expectation and Variance [closed]

A day trader buys an option on a stock that will return \$100 profit if the stock goes up today and lose \$200 if it goes down. If the trader thinks there is a 75% chance that the stock will go up, ...
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22 views

Examining median of n Bernoulli trials

During my preparation for Statistics exam I ecountered an interesting problem: We have $n$ fair coins and we flip each one until we get first head. Let $X_i$ be the number of throws until we get ...
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31 views

Application of dominated convergence theorem to conditional expectation

Suppose we have the following sequence of random variables such that \begin{align} Z_a&=E^2[V^2|aV+W]\\ \lim_{a \to 0} Z_a&= E^2[V^2] \quad (a.s.) \end{align} where $V$ and $W$ are ...
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1answer
63 views

Expected value of $X+Y$

Why does $E[X+Y]$ equal the following?: $$\mathbb{E}[X+Y] = \sum_{x,y}(x+y)P(X=x,Y=y)$$ I dont understand why its a joint probability instead of $P(X=x+Y=y)$. Can somebody please make it clear why ...
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1answer
38 views

The expected range covered by a random walk

The question that I have been struggling with lately is: If we have a one-dimensional random walk of length $n$ (consisting of $n$ steps) with discrete steps $1$ and $-1$, with probabilities of ...
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32 views

Fubini's Theorem and expectation of random variables

I have a question regarding the application of the Fubini's Theorem to the expectation of the product of two random variables. Let $X,Y$ be two random variables defined on the probability space ...
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0answers
57 views

How to find the expected value of $f(x)=\frac{1}{2\sigma}\exp\left(-\frac{|x|}{\sigma}\right)$

How to find the expected value of x with pdf $f(x)=\frac{1}{2\sigma}\exp\left(-\frac{|x|}{\sigma}\right)$, where x takes all real numbers. The original question asks to find the Method of Moment ...
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1answer
29 views

Integral estimate for positive part

I'm somewhat stuck on understanding, what seems to be a kind of elementary estimate. Let $X\geq 0$ with $\mathbb{E}(X)=1$. $f:\mathbb{R}^+ \to \mathbb{R} $ and $f(x)\geq-c>-1$. Let $A$ be some ...
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1answer
35 views

Application of Doob's inequality

Suppose that $X_n$ is a martingale with $X_0 = 0$ and $EX^2_n < \infty$. Show that $$P\left(\max_{1\leq m \leq n} {X_m} \geq \lambda\right) \leq \frac{EX^2_n}{EX^2_n+\lambda^2}$$ by using ...
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20 views

Different definitions of expectations: which types of integral do they involve?

Consider a random variable $X: \Omega \rightarrow \mathbb{R}$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. The probability space induced by $X$ is $(\mathbb{R}, ...
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28 views

How to find $\limsup_{a \to 0} E[V | aV+W]$

How to find \begin{align} \limsup_{a \to 0} E[V | aV+W] \end{align} where $V$ and $W$ are independent. We can assume that $E[V^2], E[W^2] <\infty$. I think the following must \begin{align} ...
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1answer
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Linearity of Expectation = Hat-Check Problem

I was reading up on the linearity of expectations on http://www.geeksforgeeks.org/linearity-of-expectation/. One of the problems given and explained was the hat-check problem: Let there be group of n ...
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Existence and non-singularity of the Fisher information matrix

Consider a random vector $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X: \Omega \rightarrow \mathbb{R}^k$. Suppose $X$ has probability density $p_{\theta_0}$ with respect ...
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1answer
23 views

Bound on variance of bounded random variable

For a bounded random variable $X \in [a,b]$, we know $\operatorname{Var}(X) \le (b-a)^2/4$, see for example this answer. I am trying to give an alternate proof using symmetrization. If $Y$ is an ...
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2answers
33 views

Find the first and second moments of a distribution of order statistics?

I'm not totally sure how to even word this question, but I need to find the first and second moments of two variables, $M$ and $N$ as defined by: $M=\min(X_1,X_2,\dots,X_n)$ and ...
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1answer
26 views

Show the equivalency between the two expected values?

Assume $X$ and $Y$ are non-negative integers. They are random variables. Show that $E[X]=\sum_{n>o} P[X\geqslant n]$ Show also that $E[X*Y]=\sum_{n=1}^{\inf}\sum_{m=1}^{\inf}P[(X\geqslant n) \cap ...
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47 views

How do you show $XY$ has finite expectation?

Let $X$ and $Y$ be a random variables with finite variance, so that expectation of $X^2$ is finite and so is expectation of $Y^2$
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30 views

The convergence of series of independent random variables

Let $\{a_n\}$ be a sequence of complex numbers and let $\{A_n\}$ be a non-descreasing sequence of positive numbers, tending to infinity. We assume that $\sup\limits_{n\ge 1} ...
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19 views

Finite expectation (variance) in Markov inequality and Chebyshev inequality?

Consider a random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X: \Omega \rightarrow \mathbb{R}$. (1) If $\exists$ $E_{\mathbb{P}}(X)$, then $$ ...
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2answers
61 views

Expected number of successes before first failure (Hypergeometric distribution)

Think of the following scenario: We are a group of $42$ people. I tag you, and you tag another person. This other person tags another person, etc. The "chain" of tagging stops when a person has been ...
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1answer
30 views

What is the expected value of cosine of a multivariate Gaussian?

Suppose $X \sim \mathcal{N}\left(\mu, \Sigma\right)$. How do I evaluate $\operatorname{E}\left[\cos \left(t^{T}X \right) \right] $ and $\operatorname{E}\left[\sin \left(t^{T}X\right) \right] $? Does ...
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2answers
27 views

Expected value over a grid

We have a d*d grid, each square of the grid has a probability $p_{i,j}$ of being white, otherwise the square is black. Let X the discrete random variable which gives the number of white squares. What ...
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1answer
28 views

Compute $ \mathbb{E} [W(t_1)W(t_1 + t_2)W(t_1 + t_2 + t_3)] $ when $W$ is a Brownian motion

Let $(W(t))_{t \geq 0}$ be standard Brownian motion, and let $t_1, t_2, t_3 \in \mathbb{R}_{> 0}$ with $t_1 < t_2 < t_3$ be arbitrary. Compute: $$ \mathbb{E} [W(t_1) * W(t_1 + t_2) * ...
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88 views

Since $\mathbb{E}[X]$ is defined as the integral from $0$ to infinity of $S(x)$, what do you do when $-1<x<1$?

I have the PDF $f(x)=\frac{3}{4}(1-x^2)\mathbf 1_{-1<x<1}$ and accordingly the CDF $$F(x)=\begin{cases}0, &\phantom{-}x\le -1\\\frac{3}{4}x-\frac{1}{4}x^3+\frac12, & -1<x<1 \\1, ...
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206 views

Erasing numbers from the front of the row

Numbers $1,2,\ldots,k$ are written in this order in a row. For $i=1,\ldots,k$, in the $i$th step, a random variable $V_i$ is drawn uniformly from the interval $[0,2i]$. If $V_i$ is greater than the ...
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1answer
31 views

The correct way to take expectation of a r.v. which isn't finite a.e.

Given a r.v. $\tau$ and knowing that $ \Bbb{P}(\tau = \infty )> c > 0 $ 1) I know that $\Bbb{E}(\tau) = \infty $ because $ \Bbb{E}(\tau) =\Bbb{E}(\tau ;\{\tau =+\infty \})+\Bbb{E}(\tau ;\{\tau ...
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0answers
17 views

Expectation of the matrix product $Q A (Q A)^h$

Let $Q$, of dimension $m \times n$, be a matrix with i.i.d. complex Gaussian elements; we suppose that these elements have zero mean and have the same variance $= v$. We define $A$ as an $n \times k$ ...
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0answers
9 views

How to define a likelihood function for an EM algorithm

Assuming $A$ a set of vectors from a normal distribution, and $X$ a projection matrix and $B$ a set of projected vectors of $A$ using $X$: $B=A*X$ Using an EM approach and by initializing X from ...
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1answer
34 views

Creating a tree from a permutation

I'm stuck on this linearity of expectation problem. My final answer didn't make sense I checked through my calculations a few times. I think I am solving this wrong. Suppose we have a permutation ...