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

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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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2answers
36 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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28 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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2answers
39 views

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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1answer
47 views

Probability and expectation proof

Let X have PDF f(X) and let a,b ∈ R. Show that E[aX + b] = aE[X] + b. I am little confused how do you prove it? Isn't it just the regular proof that the expectation of b is just b?
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64 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 ...
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8 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...
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17 views

Expectation of truncated normal X conditional on truncated normal Y

I am trying to derive: $E(X|a \leq Y \leq b)$ where $c \leq X \leq d $, $X$ and $Y$ are (doubly truncated) Gaussians with the same mean and different variance, and $a < c < d < b$ are the ...
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27 views

Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen

Suppose I have a set of 100 integers. I randomly choose 10 of those, make a note of which ones I selected, and repeat the process. What is the expected number of times this process must be repeated ...
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36 views

Expected time until reaching absorbing state of Markov chain

I currently try to model nucleation as an absorbing Markov chain. I have an idea how to do that but, however, I cannot convince myself that it is correct. The state space consists of the number of ...
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31 views

An application of Jensen's Inequality for dependent random variables

Consider dependent and positive valued random variables $A,B$ and $X$. I want to prove that \begin{equation} E[X^2 A] E[B] \ge E[X A] E[X B]. \end{equation} If $A$ and $B$ were scalars, above would ...
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60 views

Prove the series converges a.s in Probability

I have an article as follows Why are they enough to prove that $ \sum_{n=1}^\infty \dfrac{X_n \textbf{1}_{\{|b_n|< |X_n|\}}}{b_n} $ converges almost surely? I want to know why must prove $ ...
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2answers
67 views

Expected number of coin tosses [on hold]

Consider a perfect coin with two sides (A and B) each equiprobable $P(A)=P(B)=P(\bar{A})=\frac{1}{2}$. What's the average number of tosses to get (AA) and (AB)? (all tosses independent) Closed form ...
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79 views

Prove expectation finite

Let $ \{b_n\} $ be a sequence of non-zero complex numbers. We have $ N(t)=\#\{n \geq 1:|b_{n}|\leq t\} $, $ \displaystyle\limsup_{t\rightarrow\infty} N(t)/t^p<\infty (1\leq p <2)$, for $ X_1\in ...
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1answer
20 views

Indicator Functions with Random Variables

Let $E$ be an event and $Y$ a random variable. What exactly is meant by $\mathrm E[\mathbf 1_E \mathbf 1_{Y\in B}]$? I have two guesses, the first is that $\mathbf 1_{Y\in B}$ is an indicator random ...
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1answer
18 views

expected squared prediction error derivation

I'm having a hard time deriving the formula on page 223 of Hastie et al. for the expected squared prediction error: Here are my first steps: $$ Err = E[(Y-\hat f(x))^2] = \\ E[(Y -f(x) +f(x) ...
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40 views

Capped Probability and Expected Payoff

You keep flipping a coin until you get a head. You are paid 2^(# of flips) dollars. Suppose that if you make more than 210 dollars, you will only receive 210 dollars. What is the expected payoff of ...
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28 views

What is the expected value of the absolute value of a Wiener Process?

I am trying to show that the with a Wiener Process $w(t)$, then $\mathbb{E}[|w(t_1)w(t_2)|] = (\frac{2a}{\pi}) \sqrt{t_1 \cdot t_2} (\cos \theta + \theta \sin \theta)$, given $\sin \theta = ...
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59 views

Needs an explanation on why I obtain this covariance matrix

Let's say $n$ is an even integer. I'm playing with a column vector $\mathbf{v}$ which must satisfy the following three requirements: It's a length-$n$ vector of +1s and -1s. It has the same number ...
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56 views

Is $E[Z E[Z^2\mid Y] ]$ positive or negative?

Let $Y=X+Z$ where $X$ and $Z$ are independent, zero mean, finite variance r.v. Moreover, $Z$ is Gaussian. Is there are way to say wether \begin{align*} E[Z \ E[Z^2 \mid Y] ] \end{align*} is positive ...
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31 views

Expected value of discrete uniform variable

I have a question regarding linear combinations/transformations in statistics. I'm quite sure the answer is relatively easy, but I can't seem to find a solution that corresponds to my solution manual. ...
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1answer
61 views

Card Distribution: Expected Value

Percy shuffles a standard $52$-card deck and starts turning over cards one at a time, stopping as soon as the first spade is revealed. What is the expected number of cards that Percy turns over ...
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43 views

Expectation on second moment which involves linearity

I have a small problem regarding to expectation on second moment. It would be lovely if you guys can give me a hand. The amount of a claim that a car insurance company pays out follows an ...
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32 views

Expectation of $\min(X, c)$ for $X$ truncated r.v. and $c$ constant

I have a random variable $X$ and a constant $c\geq 0$. I define the r.v. $Y = \min(X, c)$ and I want to calculate $E[Y]$. I have seen different posts on similar topics, so I am trying to pull all ...
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1answer
26 views

Can I use Birkhoff's Ergodic Theorem for this problem?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
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1answer
52 views

Expectation with exponential random variable

If $X_i$, $i=1,2,3$ are independent exponential random variable with rates $\lambda_i$, find $$E[\max(X_i) \mid X_1<X_2<X_3]$$ I really did not understand this exercise, because if ...
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3answers
61 views

Hide and seek game

$A$ and $B$ go to the Senate to play a game of Hide-and-Seek. There are $100$ rooms in the Senate, and $B$ picks one of them and hides there till the game ends. $A$, at the beginning of every turn, ...
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34 views

In terms of $a, b,$ and $\theta$, what is the biased $b(\hat \theta)$?

The Statement of the Problem: Let $\{P_{\theta}: \theta \in \Theta \}$ be a statistical model. Suppose that $\hat \theta$ is an estimator for a parameter $\theta$ and $E_{\theta}(\hat \theta) = ...
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89 views

Expectation of an increasing transformation of a random variable

Suppose $X$ is real-valued random variable and $\phi$ an increasing function. An upper set is either an open or a closed right half line. Below, all expectations are assumed to exist and $I$ denotes ...
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43 views

Name when two functions are equal under integration (expectation)?

What is it called when $E[X] = E[Y]$? That is, $$\int x f(x)\,dx = \int y g(y)\,dy.$$ What I want to say is not that the expectation of $X$ is equal to that of $Y$ but rather (the equivalent ...
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18 views

Expected Number of Visits - why is $E_x[N_x]=\sum_{n \geq 1} p_{x,x}^{(n)}$

Suppose $(X_n)_{n \geq 0}$ is a discrete-time time-homogeneous Markov chain with transition probabilities $$P[X_{n+1}= y \mid X_{1}=x] = p_{x,y}^{(n)}.$$ Let $$N_x:=\sum_{n \geq 1} 1( X_n=x)$$ denote ...
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53 views

What is $E[X|Y]$ if the random variables $X$ and $Y$ are independent?

Just looking for an explanation of how the conditional expectation "$E[X|Y]$" of any two random variables $X$ and $Y$ would change if we included the condition that $X$ and $Y$ be independent. ...
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48 views

Prove that the value of an integral is negative for arbitrary distribution

Consider the integral given by $V(\Lambda) = \int_0^1 F(x) [1 - \Lambda +\Lambda F(x)] [x f(x) - 1 + F(x)] - [1-F(x)][1 + \Lambda F(x)][F(x) + f(x) x]\; dx $, where $F$ is the cdf of some ...
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54 views

Uniform continuous distribution for cycles.

Let there be $n$ people standing in a circle and holding hands with probability $p$. What is the expectation value $E(X)$ for the number of 'chains' when $p=.5$? For what $p$ is $E(X)$ largest? ...
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Expected value of a random variable $X$ with pmf $p_X(k)=(1/2)^k, k = 1,2,…,$

The Statement of the Problem: For the random variable $X$ with pmf $p_X(k)=(1/2)^k, k = 1,2,...,$ (a) Calculate $E[X]$ and $E[X(X-1)]$. (b) Use part (a) to compute Var$[X]$. Where I Am: Ok, so I ...
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52 views

Card game questions

I write down 4 numbers on four cards and you should choose one card. After seeing the card you can decide to throw away this card and pick another one without replacement. You can stop anytime you ...
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87 views

Expected Power Product of rolling a dice .

A 15 sided dice is rolled 1000 times. Let k1,k2,k3,k4,..k15 denote the number of times 1,2,3...15 appears. How can I compute the following expected value :$$E( (k_1 k_2 k_3 k_4)^5).$$ My attempts:: ...
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1answer
34 views

$E[|X(t)|]\leq K\implies E[|X(\tau)|]\leq K $?

Let $X(t)$ be a stochastic process. Assume that, for every $t\leq M$, it holds $$E[|X(t)|]\leq K, $$ for some constant $K$. Let now $\tau\in[0,M]$ be random (stopping time). Is it true that also ...
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49 views

The limit of the integral when the set is decreasing in probability to zero.

This is an exercise problem(#2 in section 3.2) from 'A course in probability theory'. If $E(\vert X \vert ) < \infty$ and $\lim_{n \to \infty} P(A_n) = 0,$ then $\lim_{n \to \infty} \int_{A_n} X \ ...
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example computing expectation

I am trying to understand the following example: A fair die is rolled, and whichever number comes up, a fair coin is then flipped that many times. Let $N$ be the outcome of the die roll, and $X$ the ...
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Expectation of the maximum as the number of random variables goes to infinity?

Suppose $v_1,v_2,...,v_n$ are $n$ i.i.d. continuous random variables with the range $[\underline v,\bar v]$, does the expectation of the maximum of the random variables $E[\max v_i, i=1,2,\dots,n]$ go ...
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Equivalent definitions for expected value of random variable

Let $(\Omega, P)$ be a probability space. One definition for the expected value of a random variable $X$ is $$E(X)=\sum_{x\in \mathbb{R}} xP(X=x).$$ The notes I am reading say that this definition ...
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Expectation of inter arrival of time of two processess

My question is related to expected inter arrival time between two processes X and Y. I have two process repeating every a and b seconds (Let's say every 2 and 3 seconds). ...
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1answer
26 views

Comparing Percentiles of 2 Samples Drawn from the Same Distribution

Suppose I have two sets of numbers: $A=\{a_1,a_2,...a_{N_1}\}$ and $B=\{b_1,b_2,...b_{N_2}\}$ with $N_1<N_2$. WLOG assume that $a_i<a_j$ for all $i<j$ and similarly for $b_i$ and $b_j$. ...
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32 views

Expected value of inter arrival times based on the rate

My question is related to expected inter arrival time between two processes X and Y. I have two process repeating every a and b seconds (Let's say every 2 and 3 seconds). ...
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1answer
14 views

Expectation of size of bootstrapped sample

Lets say we have a sample $\mathbf{X} = \{x_1, x_2, \dots, x_N\}$. We draw $N$ points from $\mathbf{X}$ with replacement (do a $\textit{bootstrap})$. What is the expectation of size of bootstrapped ...
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35 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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18 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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22 views

Expectation of geometric summation of exponentail random variables

We have $\{X_i, i = 1,2,\ldots\}$ as a sequence of independent exponentially distributed rv's with parameter $\lambda$. We also have, $Y =\sum_{i=1}^{N} X_i$. I need to prove that, $Y$ has the ...
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1answer
32 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 ...