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

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Find a tight upper bound for the expected value of a complicated function of random variables

Given $x \in R^n$ that follows a multivariate Gaussian distribution $x \sim \mathcal{N} (x|m,V) $; where $V$ is a diagonal covariance matrix. We already knew $m;V$. We construct the the following ...
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1answer
15 views

Justify $E(X|Y)=E(E(X|Z,Y)|Y)$ [duplicate]

Why $E(X|Y)=E(E(X|Z,Y)|Y)$? I know that $E(U)=E(E(U|V))$. So, $E(U|W)$ should be $E(E(U|V,W))$. But the latter expression is free from $W$ so it is not possible. I can get a intuitive idea that ...
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2answers
15 views

How to use independence to simplify $E\left\{\sum\limits_{i=1}^n (Y_i-\mu ) \right\}^2$

I don't know how to get the second line from the first line in the following: In the above case, $Y=(y_1, \dots , y_n)^T$ is a random sample from $N(\mu,\sigma^2)$. My trouble is in simplifying $ ...
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1answer
20 views

Conditional expectation w.r.t measure and push-forward measure

I have been introduce to the theory of conditional expectation. My question is simple. I wish to know the similarities between two quantities. My book talks about the existence of $E(Y|X=x)$ and ...
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4k views

Would you ever stop rolling the die?

You have a six-sided die. You keep a cumulative total of your dice rolls. (E.g. if you roll a 3, then a 5, then a 2, your cumulative total is 10.) If your cumulative total is ever equal to a perfect ...
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1answer
25 views

Urn problem: probability of drawing balls of k unique colors

Given an urn with $N$ balls in $K$ colors, divided evenly (so $N$ $mod$ $K$ = $0$). What is the probability that I draw $k$ different colors if I do $n$ draws without replacement? And, more general, ...
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1answer
56 views

Expecation of sum of random +-1 vars

Let $X_i$ and $Y_i$ be i.i.d. random variables taking on $1$ and $-1$ with probability $\frac12$. Let $C_n = \frac 1n\sum_{i=1}^n X_i Y_i$. What is the expected value of $|C_n|$? EDIT: It looks to ...
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1answer
12 views

Expected amount of different diced numbers

I roll n regular, d-sided dices. What is the expected amount of different numbers on the dices? Or equivalently: What is the expected amount of numbers that have not been diced at all? Example: ...
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1answer
24 views

Probability - Random Walk Type Problem

Suppose two teams play a series of games, each producing a winner and a loser, until one time has won two more games than the other. Let G be the number of games played until this happens. Assuming ...
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1answer
20 views

Closed form expression for $\sigma$

A student I'm tutoring came to me with a problem in which he needs to find a closed-form expression in $\sigma$, $E(|Y|)$. $Y$ has a normal distribution with mean $0$ and standard deviation $\sigma$. ...
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1answer
21 views

Expectation and Variance of location scale exponential distribution

Let's suppose we have the following location scale exponential distribution: $P(X>x)={ e }^{ -C(x-{ x }_{ 0 }) }$ We are looking to determine $E(X)$ and $V(X)$ There are several ways to do it, ...
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1answer
33 views

Telescoping Sum of expectations: limsup exists but limes not necessarily

Let $X_t$ for $t \in \{0, 1, \dotsc, \}$ be a sequence of non-negative integer-valued random variables. Suppose that $$\mathbb{E}[X_t - X_{t+1} \mid X_t>0 ] \leq c \quad \text{ for some constant ...
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1answer
33 views

Convergence of $\operatorname E|X_n|^p$ when $0<p<1$

Let $0<p<1$ and $X_1,\ldots,X_n$ be random variables with finite absolute moments of order $p$. Suppose that the random variables $X_1,\ldots,X_n$ converge in mean of order $p$ to a random ...
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1answer
24 views

Find the constant in the following matrix

An atom is prepared in the angular momentum state $$C\left(\begin{array}{c}1 \\ 2\end{array}\right)$$Here $C$ is a constant. This has benn written in the $S_z$ basis. a)Find C b)Work out ...
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16 views

Expectation of a process with stochastic volatility

I would like to compute the conditional expectation of a stochastic process with stochastic volatility. The model is similar to Heston model except here the drift is not constant but an independent ...
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1answer
29 views

How to show the following about expectations

If $X$ is $\mathcal{M}_1$ measurable on a probability space, $\mathcal{M}_2\subseteq\mathcal{M}_1$ and $Y$ is $\mathcal{M}_2$ measurable and they satisfy $E(XI_A)=E(YI_A)$ for every ...
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Expectation of power function in Gamma Distribution?

If X~Gamma(\alpha,\beta), does the expectation of power function, E[X^t], t>0, have a closed-form solution? I know that when t is a positive integer (natural number), E[x^t] can be calculated by using ...
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3answers
56 views

How to comprehend $E(X) = \int_0^\infty {P(X > x)dx} $ and $E(X) = \sum\limits_{n = 1}^\infty {P\{ X \ge n\} }$ for positive variable $X$?

Suppose $X$ is an integrable, positive random variable. Then, if $X$ is arithmetic, we have $E(X) = \sum\limits_{n = 1}^\infty {P\{ X \ge n\} }$ and if $X$ is continuous, we have $E(X) = ...
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0answers
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Expectation of couples surviving after some time.

There are $2m$ persons forming $m$ couples who live together at a given time. Suppose that at some later time, the probability of each person being alive is $p$, independently of other persons. At ...
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1answer
21 views

Strategy for Unbalnaced Gamber Ruin

A gambler plays the following game: A fair coin is tossed until getting three times continuously head. When that happens the Gambler gets 20$\$$. Each round costs the gambler 1$ (even if he won the ...
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1answer
20 views

Expectation of Itö integral related to martingale property

A reference stated : "Because $I(t)$ is a martingale and $I(0) = 0$, we have $E[I(t)]=0$ for all $t\geq0$" What are the role of the martingale and initial value to determine that $E[I(t)] = 0$? ...
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Expectation of vector normalization

I have problem like this $\theta$ is from Dirichlet distribution $\theta \sim Dir(\alpha)$ Assume that $\theta$ is the vector of $K$ elements and I want to calculate $$ \mathbb{E} ...
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1answer
19 views

Expected length of shortest interval containing numbers drawn at random

A random idea: If you draw $n$ numbers uniformly at random from $[0,1]$, what is the expected length $L_n$ of the shortest interval that contains all but one of them? Clearly, we have $$L_2 ...
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2answers
47 views

Arriving at the formula for expected value of a random variable

I was trying to solve this programming problem but I can't solve it since I am not able to arrive at the right formula for it. I dont want the solution using Dynamic Programming. I just want someone ...
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0answers
34 views

If $X\geq0$ is a random variable, show that $\lim\limits_{n\to\infty}\frac1nE\left(\frac{1}{X}I\left\{X>\frac{1}{n}\right\}\right)=0$

If $X\geq0$ is a random variable then show that:$$\lim_{n\to\infty} \frac{1}{n} \cdot E\bigg(\dfrac{1}{X}I\bigg\{X>\dfrac{1}{n}\bigg\}\bigg)=0$$ A hint would be most appreciated. I have ...
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The Matching Problem - Expectation

The Matching Problem You write a stack of thank you cards for people who gave you presents for your birthday. You address all of the envelopes but before you can stuff them you are called away. A ...
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1answer
28 views

Show that a space is a probability space.

Let $X : \Omega \rightarrow \mathbb{R}$ be a non-negative random variable defined on a probability space $(\Omega, F, \mathbb{P})$ with $E[|X|]<\infty $. For $A\in F$ define: $$Q[A]=\frac{E[X ...
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1answer
67 views

Sequence of Random Variables with expectation 0.

Is it possible to construct a sequence of non-negative random variables such that $X_n \rightarrow \infty $ but $\mathbb{E}[X_n] \rightarrow 0$ ? I found one which converges to 0 with expectation ...
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1answer
35 views

Expectation of squared RV

Let $\Omega$ be the set of all permutations on the set {1,2,...,n}, equipped with the uniform measure. For a permutation $\sigma \in \Omega$ let $X(\sigma)$ denote the number of fixed points by ...
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1answer
49 views

measure-theoretic definition of expectation

Consider a random variable $X \colon\Omega \rightarrow \mathbb{R}$ for a probability space $(\Omega, \mathcal{F}, P)$. We had the following definition for the expectation: $$\mathbb{E}[X]= ...
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48 views

Math Expectations and Standard Deviations

Pizza Shop profits are $1, $3, and $2 for each sale of their small, medium and large pizzas, respectively. If these are the probability distributions Small Size: $1 - 0.40 Medium Size: $3 - 0.20 ...
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1answer
38 views

How did they apply the Tower property $E[X] = E[E[X\mid Y]]$

Iam trying to understand a proof in my book but there is one detail that i don't get, here it is: Let $C_1,C_2,...C_{j+k}$ be random varibles: $E[C_{j+k} \mid C_1,...C_{j} ] = E\Big[ E[C_{j+k} \mid ...
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2answers
35 views

2 out of 3 or 4 out of 6 free-throw problem - how does it mathematically make sense?

Question: Suppose you are shooting free throws and each shot has a 60% chance of going in (there is no "learning" effect and "depreciation" effect, all have the some probability no matter how many ...
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Regression model, find $Var(y_i-\hat y_i)$

For the model of $y_i = \beta_0 + \beta_1x_{i1} + e_i$ for $i = 1,...,n$, where $e_i \sim N(0,\sigma^2)$ Find $E(\hat y_i)$ and $Var(\hat y_i)$. Hence or otherwise, find $E(y_i-\hat y_i)$ and ...
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1answer
25 views

Sufficient condition for finite Expectation

In the lecture the lecturer wrote: "Assume that for a random variable $X$ it holds that $X \geq 0$ or $X \in L^1$. Then $\mathbb{E}[X] < \infty$." I can understand (or partially understand) why ...
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28 views

Probability space and conditioning: “simple” questions

Consider the discrete-time stochastic process $(X_t)_{t \geq 0}$, where $X_t \colon \Omega \rightarrow \mathbb{R}$. The tower property of conditional expectation tells us that ...
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1answer
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Is the integral finite if the integrand is $o(x^{-1})$?

According to theorem 2.2 in this file http://www.stat.umn.edu/geyer/old06/5101/notes/n1.pdf If $\lim_{x\to\infty} \frac{g(x)}{x^{-1}} =0$, nothing can be said about the existence of ...
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1answer
27 views

Explanation of this (probably very easy) derivation using conditional expectation

In my lecture notes there is written an equation $$\mathbb{E}[X_{t+1} \mid X_t =x ] \leq (1-\delta) x,$$ (how this equation is derived, does not really matter here). Then the next part is: ...
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0answers
27 views

How to interpret the expected value of the pdf of a random variable?

Suppose we have a random variable $X: \Omega \to \mathcal{X}$, with pdf $f_X$. It is clear that the expectation of any function of $X$, say $g(X)$, is $$E[g(X)] = \int_{x \in ...
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1answer
64 views

Is it true that:$X>Y$ implies $\mathbb Eg(X)>\mathbb Eg(Y)$ for $g$ is strictly increasing function

Suppose $X$ and $Y$ are two random variable. Let X>Y stochastically and $g$ is an strictly increasing function. Is it true that E(g(X))>E(g(Y)) strictly and how to prove it? By $X>Y$ ...
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1answer
17 views

Trouble understanding Iterated expection problem

I'm having trouble understanding iterated expectations. Whenever I try to search for solid examples online, I get proofs or theoretical solutions. I found an example, but it doesn't have a solution or ...
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1answer
35 views

Probability distribution with unbounded expectation

$\frac{1}{\pi(1+x^2)}$ is a valid probability distribution - it integrates to $1.$ If I take its expectation, $\int_{-\infty}^\infty \frac x{\pi(1+x^2)} dx$, I get an unbounded value. However, the ...
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Show that $E[g(X)u]=0$ in a standard regression model where $u = \hat{\beta}-E(\hat{\beta})$

Consider the standard regression model $y = X\beta + \epsilon$ where $y$ and $\epsilon$ are $(n \times 1)$ vectors and $X$ a $(n \times K)$ matrix. Let $\beta$ be any estimator of $\beta$. Let $u = ...
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1answer
40 views

Need help to understand the equation

Here is a small equation, which says $$E[min\{{D, Q}\}] = \int_{0}^{Q}{xf(x)dx} + \int_{Q}^{\infty}{Qf(x)dx}$$, where D is a continuous random variable denoting the demand on a specific day and f(x) ...
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0answers
14 views

Divergence of mean and quantiles of maximal difference between normal variables

Let $(N_k)_{k\geqslant1}$ be a sequence of independent standard normal stochastic variables, and let $X_n=|\max(N_1,\ldots,N_n)-\min(N_1,\ldots,N_n)|$. I'm wondering whether $\mathbb E(X_n)$ diverges. ...
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35 views

Probability Inequality Exercise

Let $n \in \mathbb{N}$ be non-negative. Show that if $E[|X|^{n}]$ is finite, then: $\lim\limits_{x\rightarrow\infty}x^{n}P(|X|\geq x) = 0$ Attempt at Solution By Markov's inequality, we have: ...
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expected value - last $k$ flips of coin are same

we flip a normal coin $n$ times. We mark $k=0.5log(n)$ and we mark the $i$'th value in $Xi$. $Y$ will be the number of times where the last $k$ flips were the same. What is $E[Y]$? I think this has ...
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0answers
25 views

Markov Chain - last $0.5log(n)$ Tosses of Coin

We toss a coin $n$ times and we mark $k=0.5log(n)$. $Y$ is the number of times where the last $k$ tosses were the same. What is $E(Y)$? I'm pretty sure I need to use Markov Chain but I'm not sure ...
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1answer
31 views

Theoretical Exercise from Ross' Probability Text

This is from Sheldon Ross' text, "A First Course in Probability": Use the following result that, for a nonnegative random variable Y, $E[Y] = \displaystyle\int\limits_{0}^{\infty}P(Y > t)dt$ to ...
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2answers
54 views

Expected Value of Subset

Say we have $n>1000$ a number that can be divided by 10. We choose randomly a subset $S$ of the numbers ${1,2...n}$ of size $n/10$. $X$ will be the amount of numbers is $S$ that can be divided by ...