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

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11 views

Estimator of expectation value for standard normal distribution

In the case of a standard normal distribution, I just read that a good estimator for E[f(x)] is $\frac{1}{M}\sum_{i=1}^M f(X_i)$ (where each $X_i$ is standard normally distributed and independent). ...
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2answers
27 views

Statistics basics

Given that $X$ has mean $a$ and variance $b$. Then $E(X^2) = a^2 + b^2$. Why is this true? Please provide a proof alongside any other relevant information. Thanks in advance.
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1answer
21 views

Finding the marginal distribution for problem with n balls.

I am trying to solve the following problem: A box contains N balls: $N_1\ white, N_2\ black,\ and\ N_3\ red\ (N = N_1 + N_2 + N_3).$ A random sample of n balls is selected from the box (without ...
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1answer
25 views

Find constants such that transformed simple symmetric random walk is martingale

Let $$S_0 :=0, \quad S_n = X_1 + ... + X_n \quad \forall n \in \mathbb{N}$$ be the simple symmetric random walk on $\mathbb{Z}$, i.e. the $X_i$ are i.i.d. with $$P[X_i = +1] = P[X_i = -1] = 1.$$ Now ...
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2answers
40 views

Finding the Expected Value with a Random Constant

Suppose $X$ is a continuous random variable with PDF: $$\begin{cases} e^{-(x-c)}\ \ \text{when }x > c \\ 0\ \quad \quad\text{when}\ x \leq c \end{cases}$$ a. Find $\mathbb{E}(X)$ b. ...
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1answer
22 views

Derivation of the Negative Hypergeometric distribution's expected value using indicator variables

I'm trying to understand how to derive the Negative Hypergeometric's expected value using indicator variables. Note, in the problem below, we are only interested in the expected value before the first ...
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2answers
73 views

Why does $\mathbb{E}[X] = \sum^\infty_{r=0}\mathbb{P}(X > r)$? [duplicate]

Why does $\mathbb{E}[X] = \sum^\infty_{r=0}\mathbb{P}(X > r)$? I understand if it was $\mathbb{E}[X] = \sum^\infty_{r=0}r\cdot\mathbb{P}(X = r)$, but for this I don't understand why.
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1answer
30 views

Definite Gaussian/exponential integral

I've been revising some quantum mechanics, and I was wondering how I would calculate a specific standard deviation. The wave function I am working with is ...
2
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2answers
49 views

n tasks assigned to n computers, what is the EX value of a computer getting 5 or more tasks?

Say a central server assigns each of n tasks uniformly and independently at random to n computers connected to it on a network. Say a computer is ‘overloaded’ if it receives 5 or more tasks. Q: ...
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1answer
27 views

Finding the mean given the probability

I'm doing some work on branching processes and would like to know where the process becomes extinct. If $X$ is the number of offspring of an individual, then the process goes extinct when ...
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1answer
21 views

Strong Markov Property for Markov Chains - Statement Verification

I suspect that my handwritten lecture notes for the Strong Markov Property are wrong. I'd appreciate corrections to them. We first define the following: A random variable $\tau$ is called a ...
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4answers
123 views

Why is $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X > k)$ true?

I'm working through a problem regarding expected values in Markov chains, and at some point it says: Recall from probability that if $X$ is a positive integer valued random variable, then ...
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0answers
5 views

An inequality involving expectation and a concave function

I have a real-valued random variable $X$ and a function defined on the real line $x \mapsto U(x)$ which is concave and strictly increasing. What I am wondering is whether the following is true. For ...
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1answer
29 views

why CCDF of x is equal to expectd value of probability of x given y

I found in an article a fact that is commonly used in scientific papers which mentions that: $\mathbb{P}(f(r)>T) = \mathbb{E}[\mathbb{P}(f(r)>T\,|\,r)]$ ($\mathbb{E}$ is with ...
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1answer
14 views

Expectation of Covariance Matrix for MVN, got answer 0 with Cov operation?

Let $x_1, \ldots x_n$ be iid realizations of a $p$-dimensional random column vector $X= (X_1, \ldots, X_p)$ such that $X \sim N_p ( \mu ,\Sigma)$. We can show that $$ \hat{\mu} = \frac{1}{n} ...
2
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1answer
70 views

Conditional Expectation of the minimum of two identical log-normal distributions

I'd like to compute the closed form mean of the minimum of two truncated log-normal distribution (on another interval than its truncation). I have: $\int_{a}^{\infty} \int_{a}^{\infty} min(v, v') \ ...
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2answers
34 views

Truncated expectation equality

can somebody please explain me how following equality works: $E [|X|^pI(|X| \leq x) ] = \int_0^x y^pdP(|X| \leq x) = x^pP(|X| > x) + p\int_0^x y^{p-1}P(|X| > y) dy$. First one should be just ...
2
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2answers
33 views

Ratio of Expected values of Boys to Girls

In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of ...
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1answer
28 views

Median of random variable X minimizes E(|X-c|) - Need help understanding a proof

Let $X$ be a continuous random variable with cdf $F(x)$. Suppose that $med(X)$ = 0, show that $\forall z: E(|X-z|) \geq E(|X|)$. Here is part of the proof I am trying to understand: I already ...
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1answer
26 views

How to compute $E[Xr / (Xr +1 - X)] $ where $X$ follows a Beta distibution?

I would like to compute $E[Xr / (Xr +1 - X)] $ where $X$ follows a Beta distribution $Beta(\alpha, \beta)$ with $\alpha, \beta > 1$, $\alpha < \beta$ and $r \in (0,1)$. This is the same as ...
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1answer
22 views

Maximize production rate - probability and expectation

A factory consists of N machines that produce products at a rate of 1 per hour. Each machine randomly chooses an hour to reset itself for maintenance, during that time the entire factory is shut down. ...
3
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0answers
23 views

covariance and expectional in proccess

Show that the process $X=(W_{\sqrt{t}}I_{(1,2)}(t))_{t \ge 0} \in \mathcal{L}_3^2$. ($W$- Wiener) Additionally calculate, for $t,s \in [1,2]$, $EX_t$ and $Cov(X_t,X_s)$ I have no idea how to start ...
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0answers
34 views

Coin Toss Until Heads Wins 2^n Dollars, n = Number of Tails - Expectation

A fair coin is tossed until it lands on heads. The player receives 2^n dollars with n being the number of tails before the heads, i.e. 1$ if first toss is H, 2 if 2nd, 4 if 3rd and so on. What is the ...
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0answers
14 views

Continuous variable calculate conditional expectation

I have this math question that I am very stuck on. Let $X$ and $Y$ be jointly continuous with density $f_{X,Y}=x+y$ if $x,y\in [0, 1]$ and zero otherwise. Let $Z=1$ if $X>Y$ and zero ...
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1answer
33 views

Expectation in reversible Markov chain

Let $X$ be a Markov chain with transition matrix: $$\mathbf{P}=\begin{pmatrix} 0 & \frac{3}{5} & \frac{2}{5} \\ \frac{3}{4} & 0 & \frac{1}{4} \\ \frac{2}{3} & \frac{1}{3} & ...
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0answers
31 views

Asymptotics of a mean of exponential terms involving Gaussians

Let $X\sim \mathcal{N}(0,I_p)$ and $\tau=\sqrt{(2-\varepsilon)\log p}$ and $\varepsilon>0$. I want to prove that for sufficiently small $\varepsilon>0$ the following holds: $$ \mathbb{E}\left[ ...
3
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2answers
29 views

Mean and Variance Geometric Brownian Motion with not constant drift and volatility

I have to derive the Geometric Brownian motion (with not constant drift and volatility), and to find the mean and variance of the solution. $\quad \left\{\begin{aligned} & d X_t = \mu(t) X_t d t ...
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0answers
27 views

How to simplify the expectation $E[e^{\theta Y}]^n$?

Let $Y$ be a uniformly distributed random variable on $[0,1]$. I am doing a problem that requires me to simplify $\frac{E[e^{\theta Y}]^n}{e^{c\theta n}}$ for some constant $c$ to get the expression ...
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4answers
49 views

How to find $E[e^{\lambda X}]$ for a random variable $X$?

Let $X$ be a random variable, how can we find the expectation $E[e^{\lambda X}]$? $e^{\lambda X}$ looks like a pdf of exponential distribution, but I have never seen how to find the expectation of a ...
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1answer
19 views

Mean and variance of the sum of cgf gamma and poisson distribution

Suppose that we have the sum of two cumulant generating function: $\log e^{m(e^t-1)} + \log(1-dt)^{-c}$, and we wish to find the expectation and variance without differentiation. I realize ...
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0answers
23 views

Average number of connected triples in a random graph G(n,p)

I need to prove that the expected number of connected triples in a random graph $G(n,p)$ is $\frac{1}{2}nc^2$ when the average degree is $c$. I know that $c = (n-1) p$. I also figured out that if you ...
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1answer
36 views

How to show $P(|X-E(X)|\leq x)=1\implies V(X)\leq x^2$

Let $X$ be a random variable with finite variance. I am trying to show if $P(|X-E(X)|\leq x)=1$ then $V(X)\leq x^2$. Could somebody please help me correct my working? $|X-E(X)|\leq ...
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1answer
45 views

Show $\mathbb{E}(\sum_{i=1}^n(X_i-(\frac{1}{n}\sum_{i=1}^{n}X_i))^2)=(n-1)\mathbb{V}(X)$

Let $X_1, X_2, \ldots$ be a sequence of i.i.d random variables with finite variance and $M_n=\frac{1}{n}\sum_{i=1}^{n}X_i$. We need to show that ...
2
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1answer
25 views

Use cdf to find expectation

I have a cdf for a $\mathbf {discrete}$ random variable, $X$, $$F_X(x)=1-(1-p)^{xn}$$ where $p\in(0,1)$, $n\in\mathbb N$, $x\in\mathbb N$ My thought is to use $$E[X]=\sum_{x=0}^\infty ...
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3answers
43 views

Archer Poisson Process problem

An archer wishes to shoot an arrow at a target. The prospective flight path of the arrow is subject to birds flying past at random times, according to a Poisson process with rate $\mu$ per second. To ...
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1answer
42 views

Proof that $E(F_X(X/\sigma))=\frac12$ for every positive $\sigma$

Define $X$ to be continuous random variable symmetric about zero with cdf $F_X$ and let $\sigma > 0$ denote a constant. Now show the following: $$ E\left[F_X\left(\frac{X}{\sigma}\right)\right] = ...
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0answers
16 views

Show the sample mean converges to minus infinity when ${X_n}$ are i.i.d. and $E(X_n^+)<\infty$ and $E(X_n{^-})=\infty$

Suppose ${X_n}$ are iid and $E(X_n^+)<\infty$ and $E(X_n{^-})=\infty$. Show that if $S_n := \sum_{j=1}^nX_n$ then $\frac{S_n}{n} \rightarrow -\infty$ almost surely. A hint in my book says to use ...
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1answer
20 views

Expectation that the last bin is empty (balls and bins questions)

Say we have N balls and K bins. Let's call Y - The number of balls in the last bin. What is E(Y) ? I don't know that way to get E(Y), I think there is a way of finding it without using indicators.
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1answer
27 views

Expected sum value of permutaion

We have a set(A) of N elements. Let's assume elements are e1,e2,e3..etc. Value of each element can be 0 or 1. Another set of N elements(set B) are given, ...
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2answers
58 views

Find the mathematical expectation [closed]

Find the expectation of $$f(x) = a(1+x)^{-(1+a)}, \quad x>0.$$ The answer given is $\frac{1}{a-1}$. I am not getting the answer. Please help.
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2answers
56 views

Random points on a sphere — expected angular distance

Suppose we randomly select $n>1$ points on a sphere (all independent and uniformly distributed). What is the expected angular distance from a point to its closest neighbor? What is the expected ...
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0answers
31 views

How to find P(Y≥k) for a Poisson distribution

The original question is P(Y=y) for k-1≥y≥0 and P(Y≥k) for x=k. And find the expected value for Y. I know that the expected value of Y is the sum of two separate parts, which is the summation of ...
2
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0answers
51 views

How to use Expectation Maximization (EM) in Item Response Theory (IRT)?

Could you give a worked example on the steps of Expectation Maximization in Item Response Theory if we use the Two Parameter Rasch Model. The student abilities are unknown and the question parameters ...
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0answers
35 views

Expected value of a product with indicator function, using brownian motion

Let $\{X(t);t\geq 0\}$ be an IID stochastic process given by: $dX(t) = \mu X (t) dt + \sigma X(t) dW(t)$ where $W(t)\sim N(0,1)$ and $\mu,\sigma>0$ are constants. I want to calculate: ...
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0answers
7 views

More understanding about $E_u[\partial_x h(x,u)]$, $u$ is a random variable

Consider the subdifferential "$\partial_x h(x,u)$", $u$ is a random variable. (Note: subdifferential is a set with the definition in subgradient method.) How to understand $$E_u[\partial_x h(x,u)]$$ ...
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1answer
14 views

Does this distribution relation have a name: $\mathbb E(X^n)=b^n \mathbb E(Z^n)$ [closed]

More precisely, if $X$ and $Z$ are distributions so that $X=bZ$, then $\mathbb E(X^n)=b^n \mathbb E(Z^n)$. I found it on this site and would like to know "where it comes from".
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1answer
30 views

$0 \leq Y \leq M$ random variable, $p > 1$. Calculate $\mathbb{E}(Y^p)$

$0 \leq Y \leq M$ random variable, $p > 1$. Show that $\mathbb{E}[Y^p] = \int_0^M py^{p-1}\mathbb{P}[Y \geq y] dy$ My attempt: $\mathbb{E}[Y^p] = \int_0^{\infty} Y d\mathbb{P} = \int_0^{M} Y ...
0
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1answer
30 views

Even moments of distribution given probability density function

Given the probability density function $f(x)$, and the $𝔼[X] = \frac{2}{\sqrt{\pi\lambda}} $, how best should I go about deducing the even moments of this distribution? $f(x) = ...
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2answers
14 views

Show that $\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$

Suppose you assign a $Beta(\alpha,\beta)$ prior distribution for $\theta$, and the you observed $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies ...
0
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1answer
23 views

Is the statement $E[y - sign(s - 0.5)] = 0 $ with $s \sim Uniform(0,1) $ and $y \in \left\lbrace -1, 1\right\rbrace$ with $Prob(y=1)=s$ corect?

Suppose $s \sim Uniform (0,1)$, $y \in \left\lbrace -1, 1 \right\rbrace$ with $Prob(y =1) = s$ and $Prob(y = -1) = 1-s$. One could think of $s$ as a signal telling how likely it is that $y$ will be ...